(4) (identity element for +) There is an element e in R such that. (5) (inverse element for +) For all a R there exists x R so that.

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1 Chapter 12 - Introduction to Rings We now study of an algebraic structure called a ring This is in some sense a minimal list of characteristics of the system of integers We will see that many other systems share these properties Definition 58 A nonempty set R is said to be a ring if in R there are two binary operations, denoted by + respectively, such that for all a, b, c R: (1) (closure for +) a + b R (2) (commutativity for +) (3) (associativity for +) (4) (identity element for +) There is an element e in R such that (5) (inverse element for +) For all a R there exists x R so that (6) (closure for ) (7) (associativity for ) (8) (distributive properties) The element e in R is called the zero of the ring is usually denoted by or 0 The element x that solves the equation is denoted by x = -a Finally, note that the distributive laws are the only means for relating additive concepts to multiplicative concepts in a ring A particular set may have multiple binary operations associated with it When one wishes to emphasize particular binary operations in a ring one writes the triple (R,+, ) rather than simply R Example 61 The integers, the rational numbers, the real numbers all form a ring under the usual operations of addition multiplication We formally denote these rings by (Z,+, ), (Q,+, ) (,+, ) (The fact" that the integers form rings actually is something we take on faith that is, it is an axiom that KEJ 65

2 (Z,+, ) is a ring That (Q,+, ) ( ring),+, ) are rings is a consequence of the axiom that (Z,+, ) is a Property 59 Let [a], [b], [c] define [a] [b] [ a + b ] [a] [b] [ a b ] Then the following are true: (1) (closure of ) [a] [b] (2) (associativity of ) [a] ([b] [c]) = ([a] [b]) [c] (3) (commutativity of ) [a] [b] = [a] [b] (4) (identity element for ) [a] + [0] = [a] (5) (inverse element for ) For each [y] Z n there exists [x] so that [y] [x] = [0] (6) (closure of ) [a] [b] (7) (associativity of ) [a] ([b] [c]) = ([a] [b]) [c] (8) (commutativity of ) [a] [b] = [b] [a] (9) (identity element for ) [a] [1] = [a] (10) (distributive property) [a] ([b] [c]) = [a] [b] [a] [c] Note that in that the element [2] has no multiplicative inverse Hence, there can be no general rule for multiplicative inverses Proof Prove as many time allows EG By properties of the integers congruence classes, [a] [b] = [ab] = [ba] = [a] [b] KEJ 66

3 Definition 60 The ring R is said to have an identity if there exists so that for all It is customary to write as or 1 We will follow this custom when there is no chance for confusion (Since R may not be commutative, it is required that we write rather than simply ) Example 62 (1) The rings (Z,+, ), (Q,+, ) (,+, ) all are rings with identity, namely, the number 1 (2) The ring (,, ) is a ring with unity, namely, [1] Example 63 For each we define We show that is a ring without unity Let Then for some It follows that both belong to So, the closure properties both hold Since 0, condition (4) holds It is clear that the opposite of a multiple of 5 is also a multiple of 5 so condition (5) is seen to be satisfied Because conditions (2), (3), (7) (8) hold for all integers, they certainly hold for the multiples of 5 Thus, is a ring However, there is no multiple of 5 with the property that for all (Note: In a ring with identity the commutativity of addition is forced by the distributive laws To see this consider (1 + 1)(a + b) = 1(a + b) + 1(a + b) = 1a + 1b + 1a + 1b = a + b + a + b (1+1)(a + b) = (1 + 1)a + (1 + 1)b = 1a + 1a + 1b + 1b = a + a + b + b KEJ 67

4 Hence, a + b + a + b = a + a + b + b b + a = a + b) Definition 61 The ring R is said to be commutative if for all Example 64 The rings (Z,+, ), (Q,+, ), (,+, ), (,, ), (,+, ) are all commutative Example 65 Let M(,2) be the collection of all 2x2 matrices with real number entries We define equality by iff Addition multiplication are defined, respectively, by The identities are From the properties of matrices studied in Linear Algebra, (M( ),+,) is a ring with identity However, since, we see that (M( ),+,) is not a commutative ring KEJ 68

5 The above example shows that two nonzero elements in the ring of matrices may have a product of zero Recall that this also happened in the ring when [2] [2] = [0] Definition 62 If a b are two nonzero elements of a ring R with, then we call a b divisors of zero In particular, a is a left zero divisor b is a right zero divisor Example 66 (1) In (M( ),+,) is a left zero divisor is a right zero divisor since (2) In the ring [2] is both a left right zero divisor since [2] [5] = [0] [5] [2] = [0] [ 0 ] (3) The rings (Z,+, ), (Q,+, ), (,+, ) all are free from zero divisors As (2) of Example 66 illustrates, in a commutative ring, every left zero divisor is also a right zero divisor conversely Hence, there is no distinction between left right zero divisors in a commutative ring We will show later that the presence of zero divisors precludes the cancellation law for multiplication ( ab = ac b = c ) Definition 63 A commutative ring with identity is called an integral domain if it has no zero divisors Example 67 (1) The rings (Z,+, ), (Q,+, ), (,+, ) are all integral domains (2) If p is a positive prime, then (,, ) is an integral domain KEJ 69

6 (3) The ring [2] is a commutative ring with unit [1] [0] that is not an integral domain Definition 64 A ring with identity is called a division ring iff for all in R there exists R with So, a division ring is a ring in which each nonzero element has a multiplicative inverse Example 68 (Q,+, ), (,+, ) are both division rings Example 69 - The Hamilton Quaternions Let HQ be the collection of all symbols where (Crudely, HQ is the collection of polynomials in 1, i, j, k with real coefficients) We define equality in HQ by iff Addition is defined componentwise: Multiplication is computed by multiplying the symbols formally collecting terms using the relations One can prove that (HQ,+,) is a division ring The identities in HQ are i + 0 j + 0 k i + 0 j + 0 k If a + b i + c j + d k 0, then KEJ 70

7 The Hamilton Quaternions were discovered in 1843 by the Irish mathematician Sir William Rowell Hamilton Initially, the Quaternions were used in physics (in the area of mechanics) The Quaternions still play a role in geometry number theory today It is worth filing away the important fact that (HQ,+,) is a division ring that is not commutative! Definition 65 A commuative division ring R is called a field Example 70 The rings (Q,+, ) (,+, ) are fields Example 71 Let be the collection of all ordered pairs of real numbers (a,b) We define equality in by (a,b) = (c,d) iff a=c b = d Addition multiplication are defined, respectively, by (a,b) + (c,d) = (a+c,b+d) (a,b)(c,d) = (ac - bd, ad + bc) We leave it as an exercise for the student to show that (,+, ) is a field In fact, this field is simply the field of complex numbers where (a,b) is associated with Definition 66 Let (R,+, ) be a ring let S be a nonempty subset of R We say S is a subring of R iff the triple (S,+, ) satisfies the eight condition of Definition 27 Example 72 (1) (Z,+, ) is a subring of (Q,+, ) (2) (5Z,+, ) is a subring of (Z,+, ) (3) Let O = {, -5, -3, -1, 1, 3, 5, } under the usual operations of addition KEJ 71

8 multiplication Then O is not a subring of Z as the sum of two odd integers fails to odd we fail to satisfy the closure property for addition hence Suppose that S is a subset of a ring (R,+, ) Then the elements of the set S must satisfy conditions (2), (3), (7), (8) of Definition 27 So, to show that a subset is a subring we must only verify that conditions (1), (4), (5), (6) are satified in Definition 27 Actually, the following theorem shows that we can get away with even less Theorem 67 Let (R,+, ) be a ring let S be a nonempty subset of R Then S is a subring of R iff the following condition hold: a, b S implies a + -b, a b S Proof ( ) Trivial (! ) Assume that whenever a, b " S we have that a + -b, a # b " S Condition (6) is clearly satisfied Since S is nonempty, there exists an element s in S By hypothesis, s + -s = 0 is in S Thus, condition (4) is satisfied Now, let t be any element in S Then, by hypothesis, 0 + -b = -b is in S Hence, condition (5) is satisfied Finally, let a b be elements in S By condition (5), -b belongs to S Then, by hypothesis, a + -(-b) = a + b belongs to S $ (Strictly speaking, we need to show that in a ring that -(-b) = b for all b This will be done in the next section) Ring Ring with Division Commutative Integral Field Identity Ring Ring Domain (Z n,%,& ) (n composite) (Z p,',& ) (p prime) KEJ 72

9 (nz,+, ) ( n > 1 ) (M(( ),+, ) (2x2 matrices) (HQ,+, ) (Hamilton Quaternions) (Z,+, ) (Q,+, ) ((,+, ) (),+, ) KEJ 73