Answer to Question #233069 in Abstract Algebra for 123

Question

An element of R is called idempotent if a ²=a. Show that set of all idempotents in a commutative ring is closed under multiplication.

Answer 1

Suppose that $a,b\in R$ are both idempotent. Let us prove that $ab$ is idempotent (which is equivalent to the fact that $ba$ is idempotent by symmetry of choice of a and b). We will verify it directly :

$(ab) \cdot (ab)=abab$, by commutativity we have $(ab)\cdot (ab)=a^2b^2$, which by idempotency of a and b gives us $(ab)\cdot(ab)=ab$, so it is idempotent. Therefore, the set of idempotent elements is closed under multiplication.