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College Algebra Exam Review 277

College Algebra Exam Review 277 - containing M are M and R...

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6.3. QUOTIENT RINGS 287 Proposition 6.3.10. (Diamond Isomorphism Theorem for Rings) Let ' W R ! R be a surjective homomorphism of rings with kernel I . Let A be a subring of R . Then ' 1 .'.A// D A C I D f a C r W a 2 A and r 2 I g . A C I is a subring of R containing I , and .A C I/=I Š '.A/ Š A=.A \ I/: Proof. Exercise 6.3.5 . n We call this the diamond isomorphism theorem because of the follow- ing diagram of subrings: A C I @ @ @ A I @ @ @ A \ I An ideal M in a ring R is called proper if M ¤ R and M ¤ f 0 g . Definition 6.3.11. An ideal M in a ring R is called maximal if M ¤ R and there are no ideals strictly between M and R ; that is, the only ideals
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Unformatted text preview: containing M are M and R . Recall that a ring is called simple if it has no ideals other than the trivial ideal f g and the whole ring; so a nonzero ring is simple precisely when f g is a maximal ideal. Proposition 6.3.12. A proper ideal M in R is maximal if, and only if, R=M is simple. Proof. Exercise 6.3.6 . n Proposition 6.3.13. A (nonzero) commutative ring R with multiplicative identity is a field if, and only if, R is simple....
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