# Ring theory learning seminar

Thursday, January 14, 2016 — 11:30 AM EST

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.

Location
MC 5403

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