# PlanetPhysics/Morita Uniqueness Theorem

The main result for Morita equivalent algebras is provided by the following proposition.

\begin{theorem}Morita theorem.

Let ${\displaystyle A}$ and ${\displaystyle B}$ be two arbitrary rings, and also let ${\displaystyle F:A-mod\to B-mod}$ be an additive, right exact functor. Then, there is a ${\displaystyle (B,A)}$-bimodule ${\displaystyle {\mathcal {Q}}}$, which is unique up to isomorphism, so that ${\displaystyle F}$ is isomorphic to the functor ${\displaystyle G}$ given by ${\displaystyle A-mod\mapsto B-mod,}$ ${\displaystyle M\mapsto Q\bigotimes {}_{A}M.}$ \end{theorem}

There are also two important and fairly straightforward corollaries of the Morita (uniqueness) theorem.

\begin{theorem} {\mathbf Corollary 1.}

Two rings, ${\displaystyle A}$ and ${\displaystyle B}$, are Morita equivalent if and only if there is an ${\displaystyle (A,B)}$-bimodule ${\displaystyle M_{b}}$ and a ${\displaystyle (B,A)}$-bimodule ${\displaystyle N_{b}}$ so that ${\displaystyle M_{B}\bigotimes {}BN_{B}\simeq A}$ as ${\displaystyle A}$-bimodules and ${\displaystyle N_{B}\bigotimes {}_{A}M_{b}\simeq B}$ as ${\displaystyle B}$-bimodules. With these assumptions, one obtains:

${\displaystyle End_{A-mod}(M_{b})=B^{op},}$ ${\displaystyle End_{B-mod}(N_{b})=A^{o}p}$. Also ${\displaystyle M_{b}}$ is projective as an ${\displaystyle A}$-module, whereas ${\displaystyle N_{B}}$ is projective as a ${\displaystyle B}$-module. \end{theorem}

Proof . All equivalences of categories are exact functors, and therefore they preserve projective objects as required by Corollary 1.

\begin{theorem}Corollary 2.

• (i). If ${\displaystyle A}$ and ${\displaystyle B}$ are Morita equivalent rings, then the corresponding categories ${\displaystyle {\mathbf {m} od-A}}$ and ${\displaystyle {\mathbf {m} od-B}}$ are also equivalent.
• (ii). Furthermore, there exists a natural equivalence of categories ${\displaystyle {\mathbf {A} -bimod}\to {\mathbf {B} -bimod}}$ which takes ${\displaystyle A}$ to ${\displaystyle B}$, of course along with their natural bimodule structures.

\end{theorem}

Proof. Let ${\displaystyle M_{b}}$ and ${\displaystyle N_{b}}$ be the bimodules already defined in Corollary 1 .

For proposition (i), one utilizes the functors $\displaystyle (âˆ’ \bigotimes{}_A M_b$ and $\displaystyle (âˆ’ \bigotimes{}_B N_b)$ to prove the equivalence of the two categories.

For the second proposition (ii), one needs to employ the functor ${\displaystyle N_{b}\bigotimes {}_{A}-\bigotimes {}_{A}M_{b}:{\mathbf {A} -bimod}\longrightarrow {\mathbf {B} -bimod}}$ to prove the natural equivalence of the latter two categories.