(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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