For questions related to Morita equivalence. $2$ rings like $R, S$ are Morita equivalent (denoted by $R\approx S$) if their categories of modules are additively equivalent (denoted by ${}{R}M\approx {}{S}M$).
Two rings $R$ and $S$ (associative, with $1$) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over $R$, $R$-$Mod$, and the category of (left) modules over $S$, $S$-$Mod$. It can be shown that the left module categories $R$-$Mod$ and $S$-$Mod$ are equivalent if and only if the right module categories $Mod$-$R$ and $Mod$-$S$ are equivalent. Further it can be shown that any functor from $R$-$Mod$ to $S$-$Mod$ that yields an equivalence is automatically additive.
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