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Let a belong to a ring R with unity and suppose that an =

Chapter 13, Problem 15E

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QUESTION:

Problem 15E

Let a belong to a ring R with unity and suppose that an = 0 for some positive integer n. (Such an element is called nilpotent.) Prove that 1 – a has a multiplicative inverse in R. [Hint: Consider (1 – a)(1 + a + a2 + ··· + an–1).]

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QUESTION:

Problem 15E

Let a belong to a ring R with unity and suppose that an = 0 for some positive integer n. (Such an element is called nilpotent.) Prove that 1 – a has a multiplicative inverse in R. [Hint: Consider (1 – a)(1 + a + a2 + ··· + an–1).]

ANSWER:

Step 1 of 2

Let R be a ring with unity and .

Suppose for some positive integer n.

To prove that has a multiplicative inverse in R.

Now,

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