# Yong-Soo Jung 2. Preliminaries Throughout, A will represent a complex algebra with center Z(A), R the Jacobson radical of A. Recall that A is prime if

Size: px
Start display at page:

Download "Yong-Soo Jung 2. Preliminaries Throughout, A will represent a complex algebra with center Z(A), R the Jacobson radical of A. Recall that A is prime if"

## Transcription

1 Bull. Korean Math. Soc. 35 (1998), No. 4, pp. 659{667 ON LEFT DERIVATIONS AND DERIVATIONS OF BANACH ALGEBRAS Yong-Soo Jung Abstract. In this paper we show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is always zero. 1. Introduction In 1955 Singer and Wermer proved that the range of a continuous derivation on a commutative Banach algebra is contained in the Jacobson radical [9]. In the same paper they conectured that the assumption of continuity is not necessary. In 1988 Thomas proved the Singer-Wermer conecture [10]. Hence, derivations on Banach algebras (if everywhere dened) genuinely belongs to the non-commutative setting. The non-commutative version of the Singer-Wermer theorem is related to the commutator relation. There are various non-commutative versions of the Singer-Wermer theorem. For example, in [1] Bresar and Vukman proved that every continuous left derivation on a Banach algebra A maps A into its Jacobson radical. Also they proved that every left derivation on a semiprime ring is a derivation which maps into its center. The main purpose of this paper is to show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is always zero. Using this main result, we also show some results of derivations and left derivations. Received September 26, Mathematics Subect Classication: 46H05, 46H20. Key words and phrases: derivation, left derivation, Jacobson radical, prime radical, semiprime, semisimple.

2 Yong-Soo Jung 2. Preliminaries Throughout, A will represent a complex algebra with center Z(A), R the Jacobson radical of A. Recall that A is prime if xay 0 implies x 0or y 0,and A is semiprime if xax 0implies x 0. Alinear mapping D : A! A is called a derivation if D(xy) xd(y)d(x)y (x; y 2 A). A linear mapping D : A! A is called a left derivation if D(xy) xd(y)yd(x) (x; y 2 A). Let T be a linear mapping from a Banach space into a Banach space Y. Then the separating space of T is dened as S(T )fy2y : there exists x k! 0 in with T (x k )! yg; and T is continuous if and only if S(T )f0g(see [8]). N will denote the set of all natural numbers. 3. The results Definition 3.1. Let A be a Banach algebra. A closed 2-sided ideal J of A is a separating ideal if for each sequence fa n g in A, there exists m 2 N such that (Ja n :::a 1 )(Ja m :::a 1 ) for all n m. By Stability Lemma [3] it is easy to see that every derivation on a Banach algebra has a separating space which is a separating ideal. The following lemma is due to Cusack [2]. Lemma 3.2. Let A be a Banach algebra, and P a minimal prime ideal of A such that J 6 P, where J is a separating ideal of A. Then P is closed. The following lemma can be referred to [5]. Lemma 3.3. Let D be a left derivation on an algebra A. Then D n (xy) n,1 r0 n, 1 holds for all x; y 2 A. r [D r (x)d n,r (y)d r (y)d n,r (x)](n 2 N) The following lemma is a crucial tool in proving Lemma

3 On left derivations and derivations of Banach algebras Lemma 3.4. Let D be a left derivation on an algebra A. Suppose that P is a minimal prime ideal of A such that [D r (x);y] 2 P for all x; y 2 A and r 2 N, where [u; v] denotes the commutator uv, vu. Then D(P ) P. Proof. We shall prove that the ideal P 0 fa 2 P : D k (a) 2 P for all k 2 Ng is prime again. Since D(P 0 ) P 0, minimality of P therefore yields D(P ) P. Let P 0 6 f0g. Take a; b 2 A such that a 62 P 0 but axb 2 P 0 for all x 2 A. Choose n 2 N 0 ( N [f0g) with the property D n (a) 62 P and D m (a) 2 P for all m 2 N 0 ; m<n. Wehave to prove by induction that D k (b) 2 P for all k 2 N 0. Using Lemma 3.3, we have (1) (2) (3) D nk (axb) [D (a)d nk, (xb)d (xb)d nk, (a)] 0 D (a)d nk, (xb) 0 D (xb)d nk, (a) 0 n,1 D (a)d nk, (xb) 0 D n (a)d k (xb) n D (a)d nk, (xb) n1 661

4 (4) (5) (6) Yong-Soo Jung k,1 D (xb)d nk, (a) 0 D k (xb)d n (a) k D (xb)d nk, (a) k1 By assumption, the left-hand side always belongs to P. Assume that k 0. Since (1), (6) lie in P and (2), (3), (4) disappear, it follows that D n (a)xb 2 P for all x 2 A by the hypothesis of the lemma [D r (x);y]2 P for all r 2 N, which implies that b 2 P. Now suppose that k 1. Then (1) belongs to P since D (a) 2 P for all n,1. An application of Lemma 3.3 to (3) yields D (a)d nk, (xb) n1 D (a) n1 " nk,,1 # n k,, 1 (D i (x)d nk,,i (b)d i (b)d nk,,i (x)) ; i i0 which belongs to P since D i (b) 2 P and D nk,,i (b) 2 P for 0 i n k, k, 1by the induction hypothesis. Also another application of Lemma 3.3 to (4) yields k,1 D (xb)d nk, (a) 0 k,1 0 ",1 #, 1 (D i (x)d,i (b)d i (b)d,i (x)) D nk, (a); i i0 662

5 On left derivations and derivations of Banach algebras which belongs to P since D i (b) 2 P and D,i (b) 2 P for 0 i k, 1 by the induction hypothesis. Finally, (6) belongs to P since D nk, (a) 2 P for n k, n, 1. Hence we have D n (a)d k (xb) D k (xb)d n (a) 2 P: n k The assumption of the lemma [D r (x);y]2p for all x; y 2 A and r 2 N gives us D n (a)d k (xb) 2 P: n k Thus we obtain D n (a)d k (xb) 2 P. But D n (a)d k (xb) D n (a)[xd k (b)bd k (x) k,1 k, 1 (D i (x)d k,i (b)d i (b)d k,i (x))]: i i1 By the induction hypothesis we have D i (b) 2 P and D k,i (b) 2 P. Consequently, we see that D n (a)xd k (b) 2 P for all x 2 A. Since P is a prime ideal, it follows that D k (b) 2 P. In case P 0 f0g, we take a; b 2 A such that a 6 0 but axb 0 for all x 2 A. The remainder follows the same fashion as in case P 0 6 f0g. Then we obtain D k (b) 2 P for all k 2 N 0, and hence b 0. Wecomplete the proof. Lemma 3.5. Let D be a left derivation on a Banach algebra A with radical R. Suppose that the following conditions are satised: (1) [D n (x);y]2l for all x; y 2 A and n 2 N; (2) S(D) Z(A); where S(D) is the separating space of the left derivation D and L is the prime radical of A. Then D(A) R. 663

6 Yong-Soo Jung Proof. Let Q be any primitive ideal of A. Using Zorn's lemma, we nd a minimal prime ideal P contained in Q, and hence D(P ) P by condition (1) and Lemma 3.4. Suppose rst that P is closed. Then we can dene a left derivation D : AP! AP by D(x P ) D(x) P(x2A):Since AP is prime, Bresar and Vukman's theorem [1] implies that D 0 or AP is commutative. In the second case, D(AP ) is contained in the Jacobson radical of AP by [9] whence, in both cases, D(AP ) QP. Consequently we see that D(A) Q. Observe that S(D) is a separating ideal of A by condition (2). If P is not closed, then we see that S(D) P by Lemma 3.2. Denoting : A! A P the canonical epimorphism, we have, by [ 8, Chap. 1], S( D) (S(D)) f0g whence D is continuous. As a result, ( D)( P ) f0g, that is, D( P ) P. Hence we can also dene a continuous left derivation e D : A P! A P by e D(x P ) D(x) P (x2a). Then we see that e D(A P )iscontained in the Jacobson radical of A P by [1, Theorem 2.1], and hence e D(A P ) Q P. So we obtain that D(A) Q. It follows that D(A) Q for every primitive ideal Q, that is, D(A) R. Now we prove our main result. Theorem 3.6. Let D be a left derivation on a semiprime Banach algebra A with radical R. Then D is a derivation such that D(A) Z(A) \ R. Proof. Note that D is a derivation such that D(A) Z(A) [1, Proposition 1.6]. Since D(Z(A)) Z(A), we obtain D n (A) Z(A) for all n 2 N. Also we see that S(D) Z(A) since Z(A) is a closed subalgebra of A, Therefore, by Lemma 3.5, we have D(A) R. Consequently it follows that D(A) Z(A) \ R. Corollary 3.7. Let D be a left derivation on a semisimple Banach algebra. Then D 0. Using Corollary 3.7, we can obtain the following results of derivations and a Jordan derivation. 664

7 On left derivations and derivations of Banach algebras Corollary 3.8. ([11, Theorem 3.1]) Let D be a continuous derivation on a Banach algebra A with radical R. If [D(x);y]2Rfor all x; y 2 A, then D(A) R. Proof. Since a continuous derivation leaves the Jacobson radical invariant, we may assume that A is semisimple and [D(x);y]0for all x; y 2 A. Thus Corollary 3.7 implies that D 0. Corollary 3.9. ([1, Theorem 2.2.]) Let D be a continuous Jordan derivation on a Banach algebra A with radical R. If [D(x);x] 2 R for all x 2 A, then D(A) R. Proof. By [7, Lemma 3.2], D(R) R wherefore we may assume that A is semisimple and [D(x);x]0forall x 2 A. Note that every continuous Jordan derivation on a semisimple Banach algebra is a derivation [7, Theorem 3.3]. Since [D(x);x] 0 for all x 2 A is equivalent to [D(x);y]0for all x; y 2 A by [6, Proposition 2], we see that D is a left derivation on a semisimple Banach algebra A. Hence Corollary 3.7 implies that D 0. Definition Let A and B be Banach algebras. A linear mapping T : A! B is called spectrally bounded if there is M > 0 such that r(t (x)) Mr(x) for all x 2 A. If r(t (x)) r(x) for all x 2 A, wesay that T is a spectral isometry. If r(x) 0, then x is called quasinilpotent. (Herein, r(x) lim n!1 x n 1 n denotes the spectral radius of the element x.) Observe that the canonical epimorphism : A! AR is a spectral isometry. The next theorem is a generalization of Bresar and Vukman's theorem [1, Theorem 2.1]. Theorem Let D be a left derivation on a Banach algebra A with radical R. If D n is continuous for some n 2 N, then D(A) R. Proof. Note that the quotient algebra AR is semisimple. Let x 2 A and y 2 R and observe that xd(y) D(xy),yD(x) 2 D(R)R. This shows that D(R)R is a left ideal of A, hence (D(R)) is a left ideal of AR. A simple modication of the proof of Lemma 2.1 in [4] shows that 665

8 Yong-Soo Jung (D m (x m )) (m!(d(x)) m ) holds for all x 2 R and m 2 N. Since D n is continuous for some n 2 N, we have, for each x 2 R and k 2 N, ((D(x))) nk 1 nk ((nk)!), 1 nk (D nk (x nk )) 1 nk ((nk)!), 1 nk D n 1 n x! 0 as k!1: This shows that (D(R)) is a quasinilpotent left ideal of AR, therefore, it is contained in the Jacobson radical of AR. Semisimplicity forces D(R) R. Thus we may assume that A is semisimple. Then it follows from Corollary 3.7 that D 0. Wenow have the nal result of this paper. Theorem Let D be a left derivation on a Banach algebra A with radical R. Then D(A) R if and only if D is spectrally bounded. Proof. One way implication is obvious, so suppose that r(d(x)) Mr(x) for some M > 0 and all x 2 A. Then we know that r(xd(y)) r(d(xy), yd(x)) r((d(xy), yd(x))) r((d(xy)), (yd(x))) r((d(xy))) r(d(xy)) Mr(xy)0 whenever y 2 R and x 2 A, whence D(R) R. Hence we may assume that A is semisimple. Now, D 0 follows directly from Corollary 3.7. References [1] M. Bresar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), [2] J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. 16 (1977), [3] N. P. Jewell and A. M. Sinclair, Epimorphisms and derivations on L 1 (0; 1) are continuous, Bull. London Math. Soc. 21 (1977),

9 On left derivations and derivations of Banach algebras [4] B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math. 91 (1969), [5] Y. S. Jung, A note of left derivations on Banach algebras, Korean J. Com. and Appl. Math. 4 (1997), [6] M. Mathieu, Where to nd the image of a derivation, Banach Center Publ. 30 (1994), [7] A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), [8] A. M. Sinclair, Automatic continuity of linear operators, vol. 21, London Math. Soc., Lecture Note Ser., [9] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), [10] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), [11] B. Yood, Continuous homomorphisms and derivations on Banach algebra, Contemp. Math. 32 (1984), Department of Mathematics, Chungnam National University, Taeon , Korea 667

### On certain functional equation in semiprime rings and standard operator algebras

CUBO A Mathematical Journal Vol.16, N ō 01, (73 80. March 2014 On certain functional equation in semiprime rings and standard operator algebras Nejc Širovnik 1 Department of Mathematics and Computer Science,

### Left Jordan derivations on Banach algebras

Iranian Journal of Mathematical Sciences and Informatics Vol. 6, No. 1 (2011), pp 1-6 Left Jordan derivations on Banach algebras A. Ebadian a and M. Eshaghi Gordji b, a Department of Mathematics, Faculty

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### ON ROUGH (m, n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS

U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013 ISSN 1223-7027 ON ROUGH m, n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS Naveed Yaqoob 1, Muhammad Aslam 1, Bijan Davvaz 2, Arsham Borumand Saeid 3 In this

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction

italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University

### RINGS WITH A POLYNOMIAL IDENTITY

RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### 26 Ideals and Quotient Rings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

### AMBIGUOUS CLASSES IN QUADRATIC FIELDS

MATHEMATICS OF COMPUTATION VOLUME, NUMBER 0 JULY 99, PAGES -0 AMBIGUOUS CLASSES IN QUADRATIC FIELDS R. A. MOLLIN Dedicated to the memory ofd. H. Lehmer Abstract. We provide sufficient conditions for the

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### 3. Prime and maximal ideals. 3.1. Definitions and Examples.

COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### Rough Semi Prime Ideals and Rough Bi-Ideals in Rings

Int Jr. of Mathematical Sciences & Applications Vol. 4, No.1, January-June 2014 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com Rough Semi Prime Ideals and Rough Bi-Ideals in

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.

### Factorization in Polynomial Rings

Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18 of Gallian s Contemporary Abstract Algebra. Most of the important

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS

ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS T. BRAND, M. JAMESON, M. MCGOWAN, AND J.D. MCKEEL Abstract. For a commutative ring R, we can form the zero-divisor graph Γ(R) or the ideal-divisor

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Some Remarks About Elementary Divisor Rings

Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 1-1-1956 Some Remarks About Elementary Divisor Rings Leonard Gillman University of Texas at

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### Suk-Geun Hwang and Jin-Woo Park

Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 471 478 A NOTE ON PARTIAL SIGN-SOLVABILITY Suk-Geun Hwang and Jin-Woo Park Abstract. In this paper we prove that if Ax = b is a partial signsolvable linear

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### Lifting abelian schemes: theorems of Serre-Tate and Grothendieck

Lifting abelian schemes: theorems of Serre-Tate and Grothendieck Gabriel Dospinescu Remark 0.1. J ai décidé d écrire ces notes en anglais, car il y a trop d accents auxquels il faut faire gaffe... 1 The

### Non-unique factorization of polynomials over residue class rings of the integers

Comm. Algebra 39(4) 2011, pp 1482 1490 Non-unique factorization of polynomials over residue class rings of the integers Christopher Frei and Sophie Frisch Abstract. We investigate non-unique factorization

### Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend

### 4 Unique Factorization and Applications

Number Theory (part 4): Unique Factorization and Applications (by Evan Dummit, 2014, v. 1.00) Contents 4 Unique Factorization and Applications 1 4.1 Integral Domains...............................................

### ON UNIQUE FACTORIZATION DOMAINS

ON UNIQUE FACTORIZATION DOMAINS JIM COYKENDALL AND WILLIAM W. SMITH Abstract. In this paper we attempt to generalize the notion of unique factorization domain in the spirit of half-factorial domain. It

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### Ultraproducts and Applications I

Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

### ON SOME CLASSES OF GOOD QUOTIENT RELATIONS

Novi Sad J. Math. Vol. 32, No. 2, 2002, 131-140 131 ON SOME CLASSES OF GOOD QUOTIENT RELATIONS Ivica Bošnjak 1, Rozália Madarász 1 Abstract. The notion of a good quotient relation has been introduced as

### INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS

INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### Nilpotent Lie and Leibniz Algebras

This article was downloaded by: [North Carolina State University] On: 03 March 2014, At: 08:05 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Irreducible Representations of Wreath Products of Association Schemes

Journal of Algebraic Combinatorics, 18, 47 52, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Irreducible Representations of Wreath Products of Association Schemes AKIHIDE HANAKI

### ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS

ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS DOV TAMARI 1. Introduction and general idea. In this note a natural problem arising in connection with so-called Birkhoff-Witt algebras of Lie

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### RINGS OF ZERO-DIVISORS

RINGS OF ZERO-DIVISORS P. M. COHN 1. Introduction. A well known theorem of algebra states that any integral domain can be embedded in a field. More generally [2, p. 39 ff. ], any commutative ring R with

### ON SUPERCYCLICITY CRITERIA. Nuha H. Hamada Business Administration College Al Ain University of Science and Technology 5-th st, Abu Dhabi, 112612, UAE

International Journal of Pure and Applied Mathematics Volume 101 No. 3 2015, 401-405 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i3.7

### Some Special Artex Spaces Over Bi-monoids

Some Special Artex Spaces Over Bi-monoids K.Muthukumaran (corresponding auther) Assistant Professor PG and Research Department Of Mathematics, Saraswathi Narayanan College, Perungudi Madurai-625022,Tamil

### 4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Strongly Principal Ideals of Rings with Involution

International Journal of Algebra, Vol. 2, 2008, no. 14, 685-700 Strongly Principal Ideals of Rings with Involution Usama A. Aburawash and Wafaa M. Fakieh Department of Mathematics, Faculty of Science Alexandria

### G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### On duality of modular G-Riesz bases and G-Riesz bases in Hilbert C*-modules

Journal of Linear and Topological Algebra Vol. 04, No. 02, 2015, 53-63 On duality of modular G-Riesz bases and G-Riesz bases in Hilbert C*-modules M. Rashidi-Kouchi Young Researchers and Elite Club Kahnooj

### The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

### Amply Fws Modules (Aflk n, Sonlu Zay f Eklenmifl Modüller)

Çankaya Üniversitesi Fen-Edebiyat Fakültesi, Journal of Arts and Sciences Say : 4 / Aral k 2005 Amply Fws Modules (Aflk n, Sonlu Zay f Eklenmifl Modüller) Gökhan B LHA * Abstract In this work amply finitely

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### COMMUTING MAPS: A SURVEY. Matej Brešar 1. INTRODUCTION

TAIWANESE JOURNAL OF MATHEMATICS Vol. 8, No. 3, pp. 361-397, September 2004 This paper is available online at http://www.math.nthu.edu.tw/tjm/ COMMUTING MAPS: A SURVEY Matej Brešar Abstract. Amapf on a

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### 5.1 Commutative rings; Integral Domains

5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

### CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

### Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### THE HOPF RING FOR COMPLEX COBORDISM 1 BY DOUGLAS C. RAVENEL AND W. STEPHEN WILSON. Communicated by Edgar Brown, Jr., May 13, 1974

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 6, November 1974 THE HOPF RING FOR COMPLEX COBORDISM 1 BY DOUGLAS C. RAVENEL AND W. STEPHEN WILSON Communicated by Edgar Brown, Jr., May

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Commutative Algebra Notes Introduction to Commutative Algebra Atiyah & Macdonald

Commutative Algebra Notes Introduction to Commutative Algebra Atiyah & Macdonald Adam Boocher 1 Rings and Ideals 1.1 Rings and Ring Homomorphisms A commutative ring A with identity is a set with two binary

### MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials

Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### On Nicely Smooth Banach Spaces

E extracta mathematicae Vol. 16, Núm. 1, 27 45 (2001) On Nicely Smooth Banach Spaces Pradipta Bandyopadhyay, Sudeshna Basu Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Calcutta 700035,

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### arxiv:0804.1733v1 [math.fa] 10 Apr 2008

PUSH-OUTS OF DERIVTIONS arxiv:0804.1733v1 [math.f] 10 pr 2008 Niels Grønbæk bstract. Let be a Banach algebra and let X be a Banach -bimodule. In studying H 1 (, X) it is often useful to extend a given

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### Introduction to finite fields

Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

### TRIPLE FIXED POINTS IN ORDERED METRIC SPACES

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197-207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON

### Applying this to X = A n K it follows that Cl(U) = 0. So we get an exact sequence

3. Divisors on toric varieties We start with computing the class group of a toric variety. Recall that the class group is the group of Weil divisors modulo linear equivalence. We denote the class group