TITLE: Minimal Ideal of a Commutative Ring with Unity
QUESTION [5 upvotes]: Can anyone help me prove this? This one is from Malik's Fundamentals of Abstract Algebra.
An ideal $I$ of a ring $R$ is called a minimal ideal if $I≠{0}$ and there does not exist any ideal $J$ of R such that ${0}≠J⊂I$. 

If $I$ is a minimal ideal of a commutative ring $R$ with $1$, show that either $I^2={0}$ or $I=eR$ for some idempotent $e∈R$.

REPLY [6 votes]: Suppose $I^2 \neq 0$, then $\exists a\in I$ such that $aI \neq 0$. Hence, $aI = I$, and so $\exists e\in I$ such that $ae = a$.
Now, $J = (e^2-e)R \subset I$ and $J\neq I$ (because if $J=I$, then $aI = a(e^2-e)R = 0$), and hence $J = 0$, whence $e$ is an idempotent and $I = eR$.