Exercise 2, § 10

Let A be a (left) Artinian ring and $\mathfrak{r}$ its radical.

a) Prove that every right ideal $\mathfrak{b}$ of A contains a minimal right ideal (consider the ideals $\mathfrak{b}\cap \mathfrak{r}^p$, and observe that if $\mathfrak{b}\mathfrak{r}= 0$, then $\mathfrak{b}$ is a right $A/\mathfrak{r}$-module).

b) Prove that the left (resp. right) socle of A is the right (resp. left) annihilator of $\mathfrak{r}$. Deduce that every minimal two-sided ideal of A is contained in the intersection of the left and right socles of A.