Exercise 1, § 3

Let A be a ring and $\mathfrak{m}$ a maximal left ideal of A. Let B be the set of elements $b$ of A such that $\mathfrak{m}b\subset \mathfrak{m}$. Prove that B is the largest subring of A containing $\mathfrak{m}$ in which $\mathfrak{m}$ is a two-sided ideal. Let S be the A-module $A_s/\mathfrak{m}, D$ its commutant, and $\pi$ the canonical mapping from $A_s$ to S. Prove that, given $b\in B$, there exists a unique element $u_b$ of End$_A(S)$ such that $u_b(\pi (a)) =\pi (ab)$ for every $a\in A$. The mapping $b\mapsto u_b$ induces an isomorphism from $B/\mathfrak{m}$ to End$_A(S)$.