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1 Piatetski Shapiro Meets Chebotarev Yıldırım AKBAL Bilkent University This is a joint work with Ahmet Muhtar Güloğlu. In 1953 Ilya Piatetski-Shapiro proved in [?] an analogo of the prime number theorem for primes of the form n c, where x max n IN : n x, n runs through positive integers and c 0 is fixed. He showed that the number ( x) of these primes not exceeding a given number x is asymptotic to 1/ c x / log x provided that c (1,12 /11) c. Since then, the admissible range of c has been extended by many authors and the result is currently knownn for c (1,2817 / 2426) (cf. [?] ). In this talk, we give an asymptotic formula for Shapiro primes lying in a specified Chebotarev class. We also apply our theorem to show that there are infinitely Shapiro primes of the form x ny. 2 2 [1] Y. Akbal, A. M. Güloğlu. Piatetski Shapiro meets Chebotarev (Submitted). [2] I.I. Piatetski-Shapiro. On the distribution of prime numbers in sequences of the form [ f( n )], (Russian) Mat. Sbornik N.S. 33 (75), (1953) c [3] J. Rivat, P. Sargos. Nombres premiers de la forme [ n ], Canad. J. Math, 53 (2001), no. 2,

2 Some Results on Universal Modules of Differential Operators Melis Tekin AKÇĠN Department Of Mathematics, Hacettepe University Let R be a commutative k-algebra where k is a field of characteristic zero. We have the following exact sequence where is defined as for and is the kernel of. Note that is generated by the set { }. Let and Here is called the universal derivation of order. The left - module is called the universal module of order derivations and is denoted by Let be a hypersurface represented by Then it is known that Besides it, let be a coordinate ring of the product of a reduced hypersurface and an afine t-space, then In this talk, we will give some recent results on Universal Modules. 15

3 Flicker s correction Arguement For The Trace Formula Of GL(2) On A Number Field Ali AYDOĞDU 1, Rukiye ÖZTÜRK 1 and Engin ÖZKAN 2 Department of Mathematics, Science Faculty, Atatürk University, Erzurum, Turkey Department of Mathemaics, Science and Art Faculty, Erzincan University, Erzincan, Turkey. The correction arguement for the weighted orbital integrals that appears in the trace formula was introduced by Flicker in [4]. We give the correction arguement for the trace formula of GL(2) over a number field. The correction arguement is to introduce a global summand which does not change the global formula. However it changes the local weighted orbital integrals at the hyberbolic terms Mathematics Subject Classification. 11F72, 22E35 Key words and phrases. Trace Formula, number field, orbital integrals, weighted orbital integrals. [1] Arthur, J. "The local behaviour of weighted orbital integrals." Duke Math. J 56.2 (1988): [2] Arthur, J. "An introduction to the trace formula." In : Arthur J, Ellwood D, Kottwitz R, editor. Harmonic analysis, the trace formula, and Shimura varieties Clay Math Proc 4, Amer Math. Soc. Providence RI 2005; [3] Borel A, and Jacquet H. "Automorphic forms and automorphic representations." Proc. Sympos. Pure Math. Vol. 33. No. part I. Amer Math. Soc. Providence RI 1979; [4] Flicker, YZ. The trace formula and base change for GL (3). Lecture Notes in Math. Vol. 114 Berlin and New York: Springer-Verlag, [5] Flicker YZ. Eisenstein series and the trace Formula for GL(2) over a function field. Documenta Mathematica 2014: 19, [6] Flicker YZ., and Kazhdan D.A. "A simple trace formula." Journal d Analyse Mathématique 50.1 (1988):

4 [7] Gelbart S, and Jacquet H. "Forms of GL (2) from the analytic point of view." Proc. Sympos. Pure Math. Vol. 33. No. part I. Amer. Math. Soc. Providence RI 1979; [8] Godement R. The spectral decomposition of cusp forms. Proc. Sympos. Pure Math., 9, Amer.Math. Soc., Providence, R.L., 1966, [9] Jacquet, H., and Langlands P.R. "Automorphic forms on GL (2)." Lect. Notes Math. Vvol. 114, Berlin and New York: Springer-Verlag, (1970). [10] Platonov V, Rapinchuk A. "Algebraic groups and number theory." Academic Press, Inc., San Diego,

5 Soft Matrix Product and Soft Cyrptosystem Emin AYGÜN, Akın Osman ATAGÜN, BüĢra KILIÇ Department of Mathematics, Erciyes University, Kayseri, Turkey Department of Mathematics, Bozok University, Yozgat, Turkey Soft set theory, proposed by Molodstov, has been regarded as an effective mathematical tool to dela with uncertainties. In this work, we define two new operations on the set of soft matrices, called inverse production and charactersitic production and give their properties. We introduce soft cryptosystem as a new cryptosystem method by using inverse production and characteristic production of soft matrices. We finally define soft encryption and soft decryption. Some applications are given G25, 15A23, 94A60, 11T71 Soft sets, Soft matrix, soft matrix products, cyrptosystem, soft cyrptosystem [1] H. Aktaş,, and N. Çağman, Soft sets and soft groups, Information Sciences (2007): [2] H. Aktaş,, and N. Çağman, Erraturn to "Soft sets and soft groups", Information Sciences 3 (2009), 338. [Inform. Sci. 177 (2007), ] [3] M. I.Ali, F.Feng, X. Liu, W. K Min and M. Shabir, On some new operations in soft set theory, Computers & Mathematics with Applications, 57 (9) (2009), [4] A.O. Atagün, and A. Sezgin. Soft substructures of rings, fields and modules, Computers and Mathematics with Applications 61 (3 ) (2011): [5] K. Atanassov, Operators over interval valued intuitionistic fuzzy set, Fuzzy sets and systems 64 (1994), [6] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986),

6 [7] D. Chen, E. C. C. Tsang, D. S Yeung, and X. Wang, The parameterization reduction of soft sets and its applications. Computers and Mathematics with Applications, 44(2002), [8] D. Chen, E. C. C. Tsang, D. S Yeung, and X. Wang, Some notes on the parameterization reduction of soft sets, in: International Conference on Machine Leraning and Cybernetics. Vol. 3, 2003, pp [9] N. Çağman, and S. Enginoğlu. Soft matrix theory and its decision making. Computers & Mathematics with Applications 59 (2010): [10] F. Feng, Y. B. Jun, and X. Zhao. Soft semirings, Computers & Mathematics with Applications 56 (2008), [11] Y. B. Jun, Soft BCK/BCI-algebras. Computers & Mathematics with Applications 56 (12) (2008), [12] Z. Kong, L. Gao, L. Wang and S. Li, The parameter reduction of soft sets and algorithm, Computers & Mathematics with Applications 56 (12) (2008), [13] D. V. Kovkov, V. M. Kolbanov, D. A. Molodtsov. Soft sets theory-based optimization. Journal of Computer and Systems Sciences International 46 (6) (2007) [14] P. K. Maji, R. Biswas, A. R. Roy. Soft set theory, Computers & Mathematics with Applications 45 (2003): [15] P. K. Maji, R. Biswas, A. R. Roy. Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001) [16] P. K. Maji, R. Biswas, A. R. Roy.An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44.8 (2002), [17] P. Majumdar, S. K. Samanta. Similarity measure of soft sets, New Mathematics and Natural Computation 4.01 (2008): [18] D. Molodstov, Soft set theory-first results, Computers and Mathematics with Applications 37 (1999), [19] D. Molodstov, The description of a dependence with the help of soft sets, Journal of Computer Science and Systems Sciences International 40 (6) (2001), [20] ] D. Molodstov, The theory of Soft Sets, URSS publishets, Moscow, 2004, (in Russian) [21] M. M. Mushrif,S. Sengupta, A. K. Ray. Texture classification using a novel, soft-set theory based classification algorithm, Lecture Notes in Computer Sciences 3851 (2006)

7 [22] Z. Pawlok and A. Skowron, Rudiments of seft sets, Information Sciences 177, (2007), [23] D. Pei, and D Miao. From soft sets to information systems.in: X. Hu. Q. Liu, A. Skowron, T.Y. Lin, R.R. Yager, B. Zhang (Eds.), Proceedings of Granular Computing Computing, vol. 2 IEEE, 2005, pp [24] R. Rivest, L. Adleman, and M. Dertouzos, On data banks and privacy homomorphisms, In Foundations of Secure Computation, 1978, [25] A. R. Roy, and P. K. Maji. A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007), [26] A. Sezgin and A. O. Atagün. On operations of soft sets, Computers and Mathematics with Applications 61 (2011), [27] A. Sezgin, A. O. Atagün, Emin Aygün, A note on soft near-rings and idealistic soft nearrings, Filomat (2011) Vol. 25.(1), [28] D. Stinson, Cryptography: theory and practice, CRC Press New Jersey 1995, 573pp. [29] Q.-M. Sun, Z-L. Zhang, and J. Liu. Soft sets and soft modules, in Gouyin Wang, Tian-rui Li, Jarzy W. Gryzymala-Busse, Duoqian Miao, Andrzej Skowron, Yiyu Yao (Eds.) Rough Sets and Knowledge Technology. RST-2008, Proceedings, Springer, 2008, pp [30] Z. Xiao, L. Chen, B. Zhong, S. Ye, Recognition for soft information based on the theory of soft sets in: J. Chen (Ed.), Proceedings of ICSSSM-05.vol.2, IEEE, 2005, PP [31] C. F. Yang, A note on soft set theory, Computers and Mathematics with Applications 56 (2008), [Comput. &Math. Appl. 45 (4-5) (203), ]. [32] X. Yang, D. Yu, J. Yang, C. Wu, Generalization of soft set theory: from crisp to fuzzy case, in: Bing-Yuan Cao (Ed.), Fuzzy Information and Engineering: Proceedings of ICFIE- 2007, in: Advances in Soft Computing, vol. 40. Springer, pp [33] L. A. Zadeh,. Fuzzy sets. Information and Control 8 (1965), [34] L. Zhou, and W. Z. Wu. On generalized intuitionistic fuzzy rough approximation operators, Information Sciences 178.(11) (2008),

8 On the dot Product graph of a commutative ring Ayman BADAWI Department of Mathematical & Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates Let A be a commutative ring with nonzero identity, be an integer, and (n times). The total dot product graph of R is the (undirected) graph with vertices { }, and two distinct vertices and are adjacent if and only if (where denote the normal dot product of and Let denote the set of all zero-divisors of R. Then the zero divisor dot product graph of R is the induced subgraph of with vertices { }. It follows that each edge (path) of the classical zero divisor graph is an edge (path) of We observe that if then is a disconnected graph and is identical to the wellknown zero- divisor graph of R in the sense of Beck-Anderson-Livingston, and hence it is connected. In this paper, we study both graphs and. For a commutative ring A and we show that is connected with diameter two (at most three) and with girth three. Among other things, for we show that is identical to the zerodivisor graph of R if and only if either and A is an integral domain or R is ringisomorphic to [1] Badawi, A., On the dot product graph of a commutative ring, to appear in Comm. Algebra (2015). [2] Badawi, A., On the annihilator graph of a commutative ring, Comm. Algebra, Vol.(42)(1), (2014), DOI: / [3] Anderson, D. F., Badawi, A., On the zero-divisor graph of a ring, Comm. Algebra (36)(2008), [4] Beck, I., Coloring of commutative rings, J. Algebra 116(1988), [5] Anderson, D. F., Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra 217(1999), [6] Anderson, D. F., Mulay, S. B., On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210(2007), [7] Lucas, T. G., The diameter of a zero-divisor graph, J. Algebra 301(2006),

9 The Lattice of Generalized L-Subgroups Dilek BAYRAK (1), Sultan YAMAK (2) Karadeniz Technical University, Trabzon, TURKEY, Karadeniz Technical University, Trabzon, TURKEY, Many studies have invastigated the lattice structure of fuzzy substructures of algebraic sets such as group, ring and module. Some important results about modularity and distributivity have been obtained in these studies. We discuss some properties of the lattices of normal -L-subgroups and obtain that the lattice of all normal -L-subgroups of a group is modular. As consequence, we obtain that the lattices of all normal -fuzzy subgroups and all normal L-subgroups of a group are modular. We characterize abelian groups by the latice of -L-subgroups of these groups. We show that a finity group G is cyclic if and only if the lattice of -L-subgroups is distributive. As consequence, we obtain that the lattices of all -fuzzy subgroups and all fuzzy subgroups of a finite cyclic group are distributive. The lattice of -fuzzy subgroups is obtained pseudocomplemented. We also show that the lattice of subgroups of G is a chain if and only if the lattice of -fuzzy subgroups of G is pseudocomplemented. Key Words: modular lattice; distributive lattice; L-subgroups; -L-subgroups -fuzzy subgroups; [1] Ajmal, Naseem. "The lattice of fuzzy normal subgroups is modular." Information sciences 83.3 (1995): [2] Ajmal, Naseem, and K. V. Thomas. "The lattices of fuzzy subgroups and fuzzy normal subgroups." Information sciences 76.1 (1994): [3] Bayrak, Dilek, and Sultan Yamak. "The lattice of generalized normal L-subgroups." Journal of Intelligent and Fuzzy Systems. (DOI: /IFS ). [4] Bhakat, S. K., and P. Das. "On the definition of a fuzzy subgroup." Fuzzy sets and systems 51.2 (1992): [5] Head, Tom. "A metatheorem for deriving fuzzy theorems from crisp versions." Fuzzy Sets and Systems 73.3 (1995): [6] Jahan, Iffat. "The lattice of L-ideals of a ring is modular." Fuzzy Sets and Systems 199 (2012):

10 [7] Jahan, Iffat. "Modularity of Ajmal for the lattices of fuzzy ideals of a ring." Iranian Journal of fuzzy systems 5.2 (2008): [8] Jun, Young-Bae, Min-Su Kang, and Chul-Hwan Park. "Fuzzy subgroups based on fuzzy points." Communications of the Korean Mathematical Society 26.3 (2011): [9] Ore, Oystein. "Structures and group theory. II." Duke Mathematical Journal 4.2 (1938): [10] Tarnauceanu, Marius. "Distributivity in lattices of fuzzy subgroups." Information Sciences (2009): [11] Yao, Bingxue. -fuzzy normal subgroups and -fuzzy quotient subgroups." Journal of Fuzzy Mathematics 13.3 (2005): [12] Yuan, Xuehai, Cheng Zhang, and Yonghong Ren. "Generalized fuzzy groups and manyvalued implications." Fuzzy sets and Systems (2003):

11 RAD SUPPLEMENTED LATTICES Çiğdem BĠÇER 1 and Celil NEBĠYEV 2 1,2 Ondokuz Mayis University, Department of Mathematics Kurupelit-Atakum-Samsun/Turkey In this work, Rad--supplemented lattices are defined and investigate some properties of these lattices. Let be a complete modular lattice and 1=. If is Rad supplemented for every then is Rad--supplemented. Key words: Small elemets in lattices, Supplemented Lattices, Radical Supplemented Lattices, Rad--supplemented lattices. [1]Calugareanu, G.,2000, Lattice Concepts of Module Theory, Kluwer Academic Publisher, Dordrecht, Boston, London. [2] Alizade, R., Toksoy E., Cofinitely Supplemented Moduler Lattices Arabian Journal for Science and Engineering, Volume 36, Issue 6, Page , [3] Çalışıcı, H., Türkmen, E., Generalized- Supplemented Modules, Algebra and Discrete Mathematics Volume 10, Number 2. pp ,

12 Coprimely Structured Modules Zehra BĠLGĠN, KürĢat Hakan ORAL and Ünsal TEKĠR Department of Mathematics, Yildiz Technical University, Esenler, İstanbul, Turkey Department of Mathematics, Marmara University, Kadıköy, İstanbul, Turkey Let R be acommutative ring with identity. In [5], the authors define strongly prime submodules of multiplication R-modules and call a multiplication R-module M strongly 0- dimensional provided each prime submodule of M is strongly prime. Then theu give some properties of such modules. A prime submodule P of a multiplication R-module M is called coprimely structured if, whenever P is coprime to each element of an arbitrary family of submodules of M, the intersection of the family is not contained in P. A multiplication R- module M is called coprimely structured provided each prime submodule of M is coprimely structured. We show that every strongly 0- dimensional module is coprimely structured and we conclude that every artinian multiplication module is coprimely structured. Then we give some characterizations of coprimely structured modules. [1] Abd E Bast, Z., Smith, P. P., (1988). Multiplication Modules. Communications in Algebra, 16(4): [2] Ali, M. M., (2008). Idempotent and Nilpotent Submodules of Multiplication Modules. Communications in Algebra, 6(412): [3] Ameri, R., (2003). On the Prime Submodules Of Multiplication Modules. International Journal of Mathematics and Mathematical Sciences, 2003(27): [4]Brewer, J., Richman, F.. (2006). Subrings of Zero-dimensional Rings. Multiplicative Ideal Theory in Commutative Algebra, Springer US, [5] Oral, K. H.. Özkirişçi, N. A.. Tekir, Ü., (2014). Strongly 0-dimensional Modules. Canadian Mathematical Bulletin, 57(1):

13 On Reciprocity Formula of Character Dedekind Sums Mümün CAN and M. Cihat DAĞLI Departments of Mathematics, Akdeniz University, Antalya, Turkey Applying character analogue of the Euler-Maclaurian summation formulato generalized Bernoulli function we obtain a relation involving an integral and character Dedekind sum. With the help of this relation we give an alternative prof for the reciprocity formula of character Dedekind sum. [1] B.C. Berndt, Character transformation formulae similar to those fort he Dedekind Etafunction, in Analytic Number Theory,Proc. Sym. Pure Math. XXIV, Amer. Math. Soc., Prodivence. R. I., (1973) [2] B.C. Berndt, Character analogues of Poisson and Euler-Maclaurin summation formulas with applications, J. Number Theory 7 (1975) [3] M. Cenkci, M. Can and V. Kurt, Degenerate and character Dedekind sums, J. Number Theory 124 (2007) [4] M. Can and V. Kurt, Character analogues of certain Hardy-Berndt sums, Int. J. Number Theory, 10 (2014),

14 p -adic Character Dedekind Sums Mehmet CENKCĠ Akdeniz University, Department of Mathematics, Antalya, 07058, Turkey Using p-adic measure theory we give an explicit representation of p-adic character Dedekind sums and their reciprocity laws. Keywords: Dedekind sums, character Dedekind sums, p-adic measure theory. MSC: 11F20 27

15 Permuting n-f-derivations on Lattices Sahin CERAN, Mustafa AġÇI, and Utku PEHLĠVAN Pamukkale University Science and Arts Faculty Depatrment of Mathematics KINIKLI Denizli /TURKEY In this paper as a generalization of permuting tri-derivation on a lattice we introduced the notion of permuting n-f-derivation of a lattice. We defined the isotone permuting n-fderivation and got some interesting results about isotoneness. We characterized the distributive and isotone lattices by permuting n-f-derivation Mathematics Subject Classification. 06B35, 06B99, 16B70. Key words and phrases. Lattice, Derivation, Permuting n-f-derivation. 28

16 Some Power Series with Rational Coefficients for Lioville Number Arguments Fatma ÇALIġKAN Istanbul University, Faculty of Science, Department of Mathematics, Vezneciler/Istanbul, Turkey, In the present talk, we discuss that some power series with rational coefficients belong to either the rational number field or the transcendental number set for Liouville number arguments under certain conditions. [1] Bugeand, Y. Approxiamation by Algebraic Numbers, Cambridge University Press, [2] Koksma, J. F. Über die Mahlersce Klasseneinteilung der transzentean Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh Math Phys 48, , [3] Lang, S. Integral Points on Curves, Institut des Hautes Etudes Scientifiques Publication Mahematiques 6, 27-43, [4] Leveque, W. J. Topics in Number Theory Vol. II (Addison-Wesley Publishing Company, 1961). [5] Long, X. X. On Mahler s Classification of p-adic Numbers, Pure Apply Math 5, 73-80, [6] Mahler, K. Zur Approximation der Exponantialfunktion und des Logarithmus I, Journal fr reine und angewandte Mathematik 166), , [7] Mahler, K. Über eine KLassen-Einteilung der p-adischen Zahlen Mathematica Leiden 3, , [8] Oryan, M.H. Über gewisse Potenzreihen deren Funktionwerte für Argümente aus der Menge der Liouvilleschen zahlen transzendent sind, İstanbul Üniv. Fen Fak. Mat Derg 49, 37-38, [9] Wirsing, E. Approximation mit Algebraischen Zahlen beschrankten Grades, J reine angew Math 206, 67-77,

17 Yusuf DANIġMAN Mevlana University, Konya, Turkey, Let be a non-archimedean field. The local Langlands conjecture describes a correspondence between the irreducible admissible representations of a reductive group and the representations of the Weil-Deligne group. In this conjecture, local factors of irreducible representations of are defined as the local factors of the corresponding representation of the Weil-Deligne group[?],[?]. Local factors of can be also defined by attaching an integral repre-sentation, which would compute the local factor and provides more information about the correspondence. For the group ) of symplectic similitudes of rank, Piatetski-Shapirodefined the local factors ( and ) by using the theory of integral representations depends on the Bessel model [?]. For the irreducible representations of induced from Borel subgroup we compute the -factors by using the Jacquet module structure. Key words: L-function, L-factor, [1] Deligne P., Les constantes des quations fonctionnelles des fonctions L, Lecture Notes in Math. Vol. 349, Springer Verlag, 1973, [2] Langlands R.P, On the functional Equation of Artin's L function, unpublished manuscript. [3] Piatetski-Shapiro I. L-functions for GSp4. Pacific J Math Olga Taussky Todd Memorial Issue 1997;

18 On Generalızed Semiperfect Rings Yılmaz Mehmet DEMĠRCĠ Sinop University, Department of Mathematics, Osmaniye Köyü Nasuhbaşoğlu Mevkii, 57000, Sinop, Turkey, Throughout all rings are associative with identity element unless otherwise stated and all modules are unitary right modules. Let R be a ring and M be an R- module, A flat cover of M is an epimorphism where F is a flat R-module and Amini et. al. called a ring R right generalized perfect (right G-perfect for short) if every right R-module has a flat cover. We introduce right generalized semi perfect rings (right G-semiperfect for short) as the rings pver which every simple right R-module has a flat cover. Since every projective module is flat, every semiperfect ring is G-semiperfect and therefore G-semiperfect rings are generalizations of semiperfect rings. We give some examples along with the following results concerning G-semiperfect rings. Theoem 0.1. Let R and S be Morita equivalent rings. If R is right G-semiperfect, then so is S. Proposition 0.1. Let R and S be G-semiperfect rings. Then, a) Every factor ring of R is G-semiperfect, b) RxS is G-semiperfect ring. Theoem 0.2. ([5], Theorem 3.8). Let R be a semilocal ring. Then R is G-semiperfect if and only if R is semiperfect. Proposition 0.2. Let R be a commutative G-semiperfect ringi Then is semiperfect for every finite number of maximal ideals,, and. Proposition 0.3. Let R be a right G-semiperfect ring and J(R) be nil. Then R is right Noetherian if and only if R is right Artinian Mathematical Subject Classification. Primary: 16L30; Secondary: 16D40. Key words and phrases. Generalized perfect ring, generalized semiperfect ring. [1] A. Amini. B. Amini M. Ershad, and H. Sharif, On Generalized Semiperfect Rings, Comm. Algebra 35 (2007),

19 [2] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer, New York, [3] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Westview Press [4]T, Y. Lam. A First Course in Noncommutative Rings, Graduate Texts in Matheatics, Springer, New York, [5] C. Lomp, On Semilocal Rings and Modules, Comm. Algebra 27(4) (1999),

20 On Skew Cyclic and Quasi-Cylic Codes Over Abdullah DERTLĠ a, Yasemin ÇENGELLENMĠġ b, ġenol EREN a (a) Ondokuz Mayıs University, Faculty of Arts and Sciences, Mathematics Department Samsun, Turkey (b) Trakya University, Faculty of Arts and Sciences, Mathematics Department Edirne, Turkey By defining the Lee distance and the Lee weight of linear codes over the ring,, we construct a Gray map which is both an isometry and a weight preserving map from to. It was shown that the Gray image of cyclic code over is quasi-cyclic codes of index and the Gray image of quasi-cyclic code over is -quasicyclic code of index. Moreover, the skew cyclic and skew quasi-cyclic codes over introduced and the Gray images of them are determined. D. Boucher, W. Geiselmann, F. Ulmer, Skew cyclic codes, Appl. Algebra, Eng. Commun Comput., Vol, No D. Boucher, P. Sole, F. Ulmer, Skew constacyclic codes over Galois rings, Advance of Mathematics of Communication, Vol., Number. D. Boucher, F. Ulmer, Coding with skew polynomial rings, Journal of Symbolic Computation, I. Siap, T. Abualrub, N. Aydın, P. Seneviratne, Skew cyclic codes of arbitrary length, Int. Journal of Information and Coding Theory,. J. F. Qian, L. N. Zhang, S. X. Zhu, -constacyclic and cyclic codes over, Applied Mathematics Letters,. J. Gao, L. Shen, F. W. Fu, Skew generalized quasi-cyclic codes over finite fields, arxiv: M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., DOI. M. Wu, Skew cyclic and quasi-cyclic codes of arbitrary length over Galois rings, International Journal of Algebra, vol,, no,. 33

21 S. Jitman, S. Ling, P. Udomkovanich, Skew constacyclic codes over finite chain rings, AIMS Journal. T. Abualrub, A. Ghrayeb, N. Aydın, I. Siap, On the construction of skew quasi-cyclic codes, IEEE Transsactions on Information Theory, Vol, No,,. T. Abualrub, P. Seneviratne, Skew codes over ring, Proceeding of the interntional Multi Conference of Eng. And Comp. Sci., IMECS, March,, Hong Kong. Y. Cengellenmis, On -cyclic codes over, Int.J. Contemp. Math. Sciences, vol,, no,. 34

22 TL-Ring Homomorphisms Ümit DENĠZ 1 1 Recep Tayyip Erdoğan University Department of Matematics, Rize, Turkey We study TL-ring homomorphism. TL-ring homomorphism is a TL-subset. In this paper we use fuzzy function definition and with this definition and with this definition we introduce TL-ring homomorphism and prove some theorems of ring homomorphisms in classical algebra. 1. Klement E. P., Mesiar R. And Pap E., Triangular Norms. Position Paper I: Basic Analytical and Algebraic Properties, Fuzzy Sets and Systems 143 (2004) Klement E. P., Mesiar R. And Pap E., Triangular Norms. Position Paper II: General Constructions and Parameterized Families, Fuzzy Sets and Systems 145 (2004) Klement E. P., Mesiar R. And Pap E., Triangular Norms. Position Paper III: Continuous t- Norms, Fuzzy Sets and Systems 145 (2004) Demirci M. and Recasens J., Fuzzy Groups, Fuzzy Functions and Fuzzy Equivalence Relations, Fuzzy Sets and Systems 144 (2004) Demirci M., Fuzzy Functions and Their Applications, Journal of Mathematical Analysis and Applications 252 (2000) Demirci M., Fundamentals of M-vague Algebra and M-vague Arithmetic Operations, Int. J. Uncertainly, Fuzziness Knowledge-Based Systems 10, 1 (2002) Sostak A. P., Fuzzy Functions and an Extension of the Category L-Top of Chang- Goguen L-Topological Spaces, Proceedings of the Ninth Prague Symposium, pp , Topology Atlas, Toronto, Wang Z.D. and Yu Y. D., TL-subrings and TL-ideals, Part 2: Generated TL-ideals, Fuzzy Sets and Systems 87 (1997) Wang Z.D. and Yu Y. D., TL-subrings and TL-ideals, Part 1: Basic concepts, Fuzzy Sets and Systems 68 (1994) Zadeh L. A., Fuzzy Sets, Information and Control, 8 (1965)

23 On the Character Degrees of Solvable Groups Temha ERKOÇ ¹ ¹ Department of Mathematics, University of Istanbul, Istanbul, Turkey A difficult problem in the character theory of finite solvable groups, known as the Isaacs-Seitz conjecture, asserts that the derived length of a finite solvable group is bounded above by the number of distinct irreducible character degrees of that group. The first result for this conjecture appeared in the paper [1]. Also in 1976, T. Berger proved that the conjecture is true for solvable groups of odd order. In this talk we will provide some affirmative answers to the conjecture. Keywords and phrases: Character degrees, derived lenght, solvable groups Mathematic Subject Classification: 20C15 [1] I.M., Isaacs, Character degrees and derived lenght of a solvable group, Canad. J. Math, 27 (1975), [2] T.R. Berger, Characters and derived length in groups of odd order, J. Algebra 1976, 39: [3] S. Garrison, On Groups with a Small Number of Character Degree,Ph.D. Thesis, University of Wisconsin, Madison, [4] I.M., Isaacs, Groups having at most three irreducible character degrees, Proc. Amer. Math. Soc., 1969, 21: [5] I.M. Isaacs, G. Knutson, Irreducible character degrees and normal subgroups, J. Algebra, 1998, 199: [6] I.M. Isaacs, Character Theory of Finite Groups, AMS Chelsea Publishing, Providence, RI, [7] M. Lewis, Derived lengths of solvable groups having five irreducible character degrees I, Algebras Represent. Theory :

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