A commutative ring with identity element

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Reference no: EM13718769

2. Let R be a commutative ring with 1 is not equal 0, zero, and let 0 is not equal e ∈ R be an idempotent element. Note that eR = {er | r ∈ R} is also a commutative ring with identity element e.

(1) If I is an ideal of R, show that eI = {ex | x ∈ I} is an ideal of the ring eR. Furthermore if I is prime in R and e is not an element of I, show eI is prime in eR.

(2) If J is an ideal of the ring eR, show that there exists a (unique) largest ideal I of R such that eI = J. Furthermore if J is prime in eR, show that I is prime in R.

[HINT: You may use Zorn's Lemma in proving (2).]

NOTE: I will attach the hand writing of the question so it will be better if you check the attachment.

Reference no: EM13718769

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