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lecture26

# lecture26 - 26 Examples of quotient rings In this lecture...

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26. Examples of quotient rings In this lecture we will consider some interesting examples of quotient rings. First we will recall the definition of a quotient ring and also define homo- morphisms and isomorphisms of rings. Definition. Let R be a commutative ring and I an ideal of R . The quotient ring R/I is the set of distinct additive cosets a + I , with addition and multipli- cation defined by ( a + I ) + ( b + I ) = ( a + b ) + I and ( a + I )( b + I ) = ab + I. Definition. Let R and S be rings. (i) A mapping ϕ : R S is called a ring homomorphism if ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) and ϕ ( ab ) = ϕ ( a ) ϕ ( b ) for all a, b R. (ii) A ring isomorphism is a bijective ring homomorphism. (iii) The rings R and S are called isomorphic if there exists a ring iso- morphism ϕ : R S . Example 1: Let R = Z and I = n Z for some n > 1. Let us show that the quotient ring R/I = Z /n Z is isomorphic to Z n (as a ring). Proof. In the course of our study of quotient groups we have already seen that Z /n Z = { 0 + n Z , 1 + n Z , . . . , ( n - 1) + n Z } as a set. Moreover, by Proposition 22.3, Z /n Z is isomorphic to Z n as a group with addition, and an explicit isomorphism is given by the map ι : Z /n Z Z n where ι ( x + n Z ) = [ x ] n ( * * * ) This means that the map ι : Z /n Z Z n given by (***) is (a) well-defined (b) bijective (c) preserves group operation (addition), that is, ι (( x + n Z ) + ( y + n Z )) = ι (( x + y ) + n Z ) for all x, y Z We claim that ι is actually a ring isomorphism. In view of (a), (b) and (c) it remains to check that ι also preserves multiplication, which can be done directly (using the definition of multiplication in both Z /n Z and Z n ): ι (( x + n Z )( y

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