Hw12 Problem 11

Problem: (+) An element $a$ in a ring $R$ is idempotent if $a^2 = a$.

1. Show that the set of all idempotent elements of a commutative ring is closed under multiplication.
2. Find all idempotents in the ring $\mathbb{Z}_4 \times \mathbb{Z}_{12}$.

Solution:

1. Let $a, b$ be idempotent elements within a commutative ring $R$. Now, $(ab)^2 = (ab)(ab)$ and since $R$ is a commutative ring, we can rearrange the terms to $(aa)(bb) = a^2b^2 = ab$. Since $(ab)^2 = ab$, it is clear that all idempotent elements of a commutative ring are closed under multiplication.
2. The idempotent elements of $\mathbb{Z}_4 \times \mathbb{Z}_{12}$ are:
• (1,1)
• (1,4)
• (1,9)