Morita equivalence

Remarks

Example: for a field $k$, the rings $k$ and $M_2(k)$ are Morita equivalent. The functor sends a vector space $V$ to $V^2 = k^2 \otimes_k V$, the column vectors with entries in $V$, on which $M_2(k)$ acts by matrix multiplication. A quasi-inverse sends a module $N$ to $e_{11} N$, where $e_{11}$ is the matrix unit. Non-example: $k$ and $k \times k$ are not Morita equivalent. In $k\text{-Mod}$ every nonzero module has a nonzero map to every other nonzero module, while the simple $k \times k$-modules $k \times 0$ and $0 \times k$ have no nonzero map between them. An equivalence of categories preserves this property. An equivalence between module categories is automatically additive and is naturally isomorphic to $P \otimes_R -$ for an $S$-$R$-bimodule $P$. Morita's theorem: $R$ and $S$ are Morita equivalent if and only if $S \cong \End_R(P)^{\mathrm{op}}$ for some progenerator $P$ of $R\text{-Mod}$, where $\End_R(P)$ acts on $P$ from the left. The definition with right modules gives the same relation. Isomorphic rings are Morita equivalent, and for commutative rings the converse holds.