Antiisomorphisms of endomorphism rings of modules close to free ones which are induced by Morita antiequivalences
Let the modules \(M_A\) and \(N_B\) be strong generators (i.e. they contain free cyclic direct summands) and let \(\alpha\colon\text{End}(M_A)\to\text{End}(N_B)\) be an antiisomorphism of their endomorphism rings. Suppose that \(M_A\) is a torsionless module (i.e. the canonical homomorphism \(M\to M^{**}\) is a monomorphism). In this situation some conditions are shown, which are equivalent to the following: 1. \(\alpha\) is induced by a Morita antiequivalence, i.e. by a category equivalence \(H\colon\text{Mod-}A^{op}\to\text{Mod-}B\). The conditions obtained are formulated using rank-1 idempotents of the rings studied. For example: 2. If \(e\in\text{End}(M_A)\) is a rank-1 idempotent (i.e. \(e^2=e\) and \(eM\) is a free cyclic module), then \((e^\alpha N)_B\) is a progenerator of \(\text{Mod-}B\); 3. There exist rank-1 idempotents \(e\in\text{End}(M_A)\) and \(u\in\text{End}(N_B)\) such that \((e^\alpha N)_B\) and \((u^{\alpha^{-1}} M)_A\) are generators.
scientific article; zbMATH DE number 1259437