Exercise 7, § 7

Let A be a ring.

a) Let $\mathfrak{a}$ be a minimal left ideal of A. Prove that we have $\mathfrak{a}^2= 0$ or $\mathfrak{a}^2=\mathfrak{a}$. In the latter case, show that there exists an idempotent $e$ in $\mathfrak{a}$ such that $\mathfrak{a}= Ae$ (if $a\in A$ is such that $\mathfrak{a}a=\mathfrak{a}$, consider the automorphism $x\mapsto xa$ of $\mathfrak{a}$).

b) Let $e$ be an idempotent in A. The ideal $Ae$ is minimal if and only if the ring $eAe$ is a field.

c) Let $\mathfrak{a}$ and $\mathfrak{b}$ be two isomorphic left ideals of A. Prove that we have $\mathfrak{a}\mathfrak{b}=\mathfrak{b}$ if $\mathfrak{a}^2=\mathfrak{a}$, and $\mathfrak{a}\mathfrak{b}= 0$ if $\mathfrak{a}^2= 0$.