An element a in a ring R is called nilpotent if there exists ann 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 at = 0and bm = 0 for some positive integers t and m. Find an nso that (a + b)n = 0. (You just need to find any n thatwill work, not the smallest!)
(c) Show that the set of nilpotent elements in a commutativering R forms a subring of R.
(d) Does this subring from the previous question contain aunity?
(e) Are the Gaussian integers from an earlier question anintegral domain? Explain your answer.
