Theorem 512 if a a mod i and b b mod i then ab a b

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Theorem 5.12 If a a 0 (mod I ) and b b 0 (mod I ) , then ab a 0 b 0 (mod I ) . Proof. If a 0 = a + x for x I and b 0 = b + y for y I , then a 0 b 0 = ab + ay + bx + xy . Since I is closed under multiplication by elements of R , we see that ay, bx, xy I , and since it is closed under addition, ay + bx + xy I . Hence, a 0 b 0 - ab I . 2 So we define multiplication on R/I as follows: for a, b R , ( a + I ) · ( b + I ) := ab + I. The previous theorem is required to show that this definition is unambiguous. It is trivial to show that if I ( R , then R/I satisfies the properties defining of a ring, using the corresponding properties for R . Note that the restriction that I ( R is necessary; otherwise R/I would consist of a single element and could not satisfy the requirement that the additive and multiplicative identities are distinct. This ring is called the quotient ring or residue class ring of R modulo I . As a matter of notation, for a R , we define [ a mod I ] := a + I , and if I = dR , we may write this simply as [ a mod d ]. If I is clear from context, we may also just write [ a ]. Example 5.14 For n > 1, the ring Z n as we have defined it is precisely the quotient ring Z /n Z . 2 Example 5.15 Let m be a monic polynomial over R with deg( m ) = ‘ > 0, and consider the quotient ring S = R [ T ] /mR [ T ]. Every element of S can be written uniquely as [ a mod m ], where a is a polynomial over R of degree less than . 2 5.4 Ring homomorphisms and isomorphisms Throughout this section, R and R 0 denote rings. Definition 5.13 A function f from R to R 0 is called a homomorphism if it is a homomorphism with respect to the underlying additive groups of R and R 0 , and if in addition, 1. f ( ab ) = f ( a ) f ( b ) for all a, b R , and 2. f (1 R ) = 1 R 0 . Moreover, if f is a bijection, then it is called an isomorphism of R with R 0 . 41
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Note that the requirement that f (1 R ) = 1 R 0 is not redundant. Some authors do not make this requirement. It is easy to see that if f is an isomorphism of R with R 0 , then the inverse function f - 1 is an isomorphism of R 0 with R . If such an isomorphism exists, we say that R is isomorphic to R 0 , and write R = R 0 . We stress that an isomorphism of R with R 0 is essentially just a “renaming” of elements. A homomorphism f from R to R 0 is also a homomorphism from the additive group of R to the additive group of R 0 . We may therefore adopt the terminology of kernel and image, as defined in § 4.4, and note that all the results of Theorem 4.17 apply as well here. In particular, f ( a ) = f ( b ) if and only if a b (mod ker( f )), and f is injective if and only if ker( f ) = { 0 R } . However, we may strengthen Theorem 4.17 as follows: Theorem 5.14 Let f : R R 0 be a homomorphism. 1. For any subring S of R , f ( S ) is a subring of R 0 . 2. For any ideal I of R , f ( I ) is an ideal of f ( R ) . 3. ker( f ) is an ideal of R . 4. For any ideal I 0 of R 0 , f - 1 ( I 0 ) is an ideal of R (and contains ker( f ) ). 5. The restriction f * of f to R * is a homomorphism from the multiplicative group R * into the multiplicative group ( R 0 ) * , and ker( f * ) = (1 R + ker( f )) R * .
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