**
Ehsaan
Hossain,
Pure
Mathematics,
University
of
Waterloo**

"Morita theory 1: Modules"

Let $\mathrm{Mod}_R$ be the category of right $R$-modules. Two rings $R,S$ are \textit{Morita equivalent}, denoted $R\sim S$, if $\mathrm{Mod}_R$ and $\mathrm{Mod}_S$ are equivalent as categories. For example $\mathbf{C}$ is Morita equivalent to $M_2(\mathbf{C})$, because any $\mathbf{C}$-vector space can double up to become an $M_2(\mathbf{C})$-module. Many properties are Morita invariant; for instance simplicity, semisimplicity, and chain conditions.

In the first half of this semester, we aim to cover Morita's fundamental theorems, which give surprisingly down-to-earth characterizations of when two rings are Morita equivalent. We'll need some basic module theory, namely projective and injective modules. We'll also need that when we do more localization later, so it'll pay off.