Example 2

Let A be a $k$-algebra and $n\geqslant 1$ an integer. We identify the matrix algebra $\mathbf{M}_n(A)$ with the endomorphism algebra of the right A-module $A^n_d$. We have seen that A and $\mathbf{M}_n(A)$ are Morita equivalent. For every left A-module M, identify $A^n_d\otimes_AM$ with $M^n$. The algebra $\mathbf{M}_n(A)$ then has a left action on $M^n$, and we have

for $a= (a_{ij})$ in $\mathbf{M}_n(A)$ and $m= (m_i)$ in $M^n$. Morita’s theorem implies the following results:

a) Every left module over the algebra $\mathbf{M}_n(A)$ is isomorphic to a module of the form $M^n$, where M is a left A-module.

b) Let M be a left A-module. The mapping $N\mapsto N^n$ is a bijection from the set of A-submodules of M to the set of $\mathbf{M}_n$(A)-submodules of $M^n$.

c) Let M and N be left A-modules. For an A-linear mapping $g: M\rightarrow N$, let $g_n$ be the mapping $(m_i)\mapsto (g(m_i))$ from $M^n$ to $N^n$. Then the mapping $g\mapsto g_n$ is a bijection from Hom$_A(M,N)$ to Hom$_{\mathbf{M}_n(A)}(M^n,N^n)$.

d) Let M be a left A-module. The module $M^n$ over the ring $\mathbf{M}_n(A)$ is indecomposable (resp. semisimple, simple, Artinian, Noetherian, finitely generated) if and only if the A-module M is.