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Problem 1. An element in a ring R is called nilpotent if x = 0 for some me Zt. Let z be anilpotent element of the commutative ring R.. (a) Prove that r is either zero or a zero divisor. (b) Prove that x is nilpotent for all r € R. (c) Prove that 1+ z is a unit in R. (d) Deduce that the sum of a nilpotent element and a unit is a unit.

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