Let R be a ring without any nonzero nilpotent elements. Show that (ara - ra)² = 0 for all r E R and for all idempotent elements a E R. Hence, show that C(R) contains all idempotent elements.

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A: To Prove: if R is a commutative ring without zero-divisors then characteristic of R is 0 or prime.

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Q: If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an…

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Q: Let R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

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Q: III. An element x of a ring R is called idempotent if x² = x. Determine all the idempotent elements…

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Q: Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition…

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