Give examples of (commutative) subrings R of C satisfying the following containm

Question

Give examples of (commutative) subrings R of C satisfying the following containments and non-containments. If it's not possible, explain why not. 1. Q subset R, R subset R (Q notequalto R notequalto R). 2. Q subset R, R nsub R (R notequalto Q). 3. Z subset R, Q nsub R (R notequalto Z). 4. R subset R, R subset C (R notequalto R notequalto C). Let R be a ring, we say that r elementof R is an idempotent if r^2 = r. Prove that if R is a ring in which every element is an idempotent, then R is in fact commutative, and satisfies r + r = 0_R for every r elementof R.

Explanation / Answer

PLEASE POST AS SEPARATE QUESTIONS AS PER CHEGG POLICY

Answer 4) Suppose that r2 = r, for every r R.

Then, (1 + 1)2 = 1 + 1, so that 1 + 1 + 1 + 1 = 1 + 1 1 + 1 = 0 R.

Hence, 1 = 1 R and, for any r R, we have r + r = r(1 + 1) = 0, so that r = r.

Now, let a, b R. Then,

(a+b)2 = a+b a2 +ab+ba+b2 = a+b ab+ba = 0 ab = ba = ba.

Thus, R is commutative.

DO THUMBS UP ^_^

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