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Let a belong to a ring R with unity and suppose that an =
Chapter 13, Problem 15E(choose chapter or problem)
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).]
Questions & Answers
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,