# Morita equivalence and Morita duality

$\def\rmod{R\text{-Mod}} \def\modr{\text{Mod-}R} \def\smod{S\text{-Mod}} \def\mods{\text{Mod-}S}$ Conventionally, rings $R$ and $S$ are Morita equivalent, if the categories of (left) $R$-modules ($\rmod$) and (left) $S$-modules ($\smod$) are equivalent categories (via some covariant equivalence functor). I have decided to look closer into Morita equivalence of rings and for that purpose, I picked up

K.Morita: Duality for modules and its applications in the theory of rings with minimum condition. Science reports of the Tokyo Kyoiku Daigaku, Section A, 6(150), 83--142

Would someone know whether this is the right paper to look at, for Morita equivalence? I see there material on (Morita) duality: If $D_1:\rmod \to \mods$ and $D_2: \mods\to \rmod$ are contravariant category equivalences, then the pair $(D_1, D_2)$ is called a (Morita) duality. It looks like the two should be definitely related, but how? Morita equivalence is a covariant functor, while Morita duality is a contravariant equivalence switching the sides. Or should I be looking at a different paper for Morita equivalence of rings?

• Your Tex didn't work. – Kevin Carlson Feb 1 '17 at 18:15
• Not clear why. Maybe the \bold command does not work, or definitions – Rado Feb 1 '17 at 18:16
• Use the site's markdown to format text. If you don't know the syntax, you can use the formatting buttons above the post to show you what to do. e.g. *italicized text* gives italicized text. – Hurkyl Feb 1 '17 at 19:49