Let E = {a + bi : a, b ∈ Z, b is even}. 1. A subring S of a ring R is called an ideal of R if sr, rs ∈ S for all s ∈ S and for all r ∈ R. Is S an ideal of Z[i]? Justify answer.

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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I=I ∩ rad R

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A: i have provided the detailed proof in next step

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A: R is ring with unity and I is ideal of R.

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Q: if A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (A B) + (AnC).

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).

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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???

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Q: Let R be the ring Z[√−5]. (a) Prove that I = (2, 1 + √−5), the ideal generated by those two…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…