Abstract Algebra: Theory and Applications · Problem 37

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · idempotents-integral-domains

Problem

An element

x

in a ring is called an idempotent if

x^2 = x

. Prove that the only idempotents in an integral domain are

0

and

1

. Find a ring with a idempotent

x

not equal to

0

1

. Hint. Suppose that

x^2 = x

and

x \neq 0

. Since

R

is an integral domain,

x = 1

. To find a nontrivial idempotent, look in

\mathbb{M}_2(\mathbb{R})