The Morita-contexts which arise in isomorphism between some subrings of the rings of endomorphisms
Let \({\mathcal R}=\text{End}(U_R)\), \({\mathcal K}=\text{End}(V_K)\) and the idempotents \(e\in{\mathcal R}\), \(f\in{\mathcal K}\) are such that \(R=e{\mathcal R}e\), \(K=f{\mathcal K}f\), \(U\cong{\mathcal R}e\), \(V\cong{\mathcal K}f\). If the isomorphism of subrings \({\mathcal R}'\overset\sigma{\underset\tau\rightleftarrows}{\mathcal K}'\) is given, where \({\mathcal R}'\subseteq{\mathcal R}\), \({\mathcal K}'\subseteq{\mathcal K}\), then a Morita context \(({_RP_K},{_KQ_R})\) can be constructed such that \(P\cong e{\mathcal R}'\tau(f)\cong\sigma(e){\mathcal K}'f\) and \(Q\cong\tau(f){\mathcal R}'e\cong f{\mathcal K}'\sigma(e)\). If the conditions \({\mathcal R}e{\mathcal R}\subseteq{\mathcal R}'\), \({\mathcal K}f{\mathcal K}\subseteq{\mathcal K}'\), \(\tau(f){\mathcal K}\tau(f)\subseteq{\mathcal R}'\), \(\sigma(e){\mathcal K}\sigma(e)\subseteq{\mathcal K}'\) are fulfilled, then in this context \(R\cong\text{End}(P_K)\), \(K\cong\text{End}(Q_R)\), \(P\cong\Hom(Q_R,R_R)\) and \(Q\cong\Hom(P_K,K_K)\). Moreover, in this case the context defines an equivalence of categories \(\text{Mod-}R\) and \(\text{Mod-}K\) if and only if \(\sigma({\mathcal R}e{\mathcal R})={\mathcal K}f{\mathcal K}\).
scientific article; zbMATH DE number 1158850