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# Rings and proofs

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1A) Let R be a commutative ring and let A = {t &#61536; R &#61560; tp = 0R} where p is a fixed element of R. Prove that if k, m &#61536; A and b &#61536; 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 &#61536; R &#61560; r = cb for some element c &#61536; R} is an ideal of R.

2A) Let R and S be rings let &#61544; : R &#8594; S be a homomorphism. Prove that if J is an ideal of S, then I = {b &#61536; R &#61560; &#61544;(b) = c for some element c &#61536; 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 &#61536; R &#61560; st = 0R} is an ideal of R.
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https://brainmass.com/math/ring-theory/rings-proofs-commutative-rings-77463

#### Solution Preview

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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