An element a in a ring R is called nilpotent if there exists an n such...

An element a in a ring R is called nilpotent if there exists an n such that aⁿ = 0.

(a) Find a non-zero nilpotent element in M2(Z).

(b) Let R be a ring and assume a, b ∈ R have a^t = 0 and b^m = 0 for some positive integers t and m. Find an n so that (a + b)ⁿ = 0. (You just need to find any n that will work, not the smallest!)

(c) Show that the set of nilpotent elements in a commutative ring R forms a subring of R.

(d) Does this subring from the previous question contain a unity?

(e) Are the Gaussian integers from an earlier question an integral domain? Explain your answer.

Expert Solution

Colby Messinger answered 7 months ago

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An element a in a ring R is called nilpotent if there exists an n such...

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