An element x in R is called nilpotent if xm= 0 for some m ∈ Z+. 1). Show that if n = akb for some integers a and b, then ab is nilpotent in Z/nZ. 2). If a ∈Z is an integer, show that the element  ̄a (a bar) ∈ Z/nZ is nilpotent if and only if every prime divisor of n is also a divisor of a. In particular, determine all the nilpotent elements of Z/72Z. 3). Let R be the ring of functions from a nonempty set X to a field F. Prove that R contains no nonzero nilpotent elements.

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