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Rings and proofs

(See attached file for full problem description with proper symbols)

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1A) Let R be a commutative ring and let A = {t  R  tp = 0R} where p is a fixed element of R. Prove that if k, m  A and b  R, then both k + m and kb are in A.

1B) Let R be a commutative ring and let b be a fixed element of R. Prove that the set B = {r  R  r = cb for some element c  R} is an ideal of R.

2A) Let R and S be rings let  : R → S be a homomorphism. Prove that if J is an ideal of S, then I = {b  R  (b) = c for some element c  J} is an ideal of R.

2B) Let R be a commutative ring and let t be a fixed element of R. Prove that the set C = {s  R  st = 0R} is an ideal of R.
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Solution Preview

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1A. Proof:
, where is a fixed element in .
If , , then we have
and . So . This implies that . Because is a commutative ring, then . Thus .

1B. Proof:
for some element . We show that is an ideal of .
First, I claim that is a ...

Solution Summary

This shows how to work with rings to complete proofs. Commutative rings are analyzed.

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