Corner replacement for Morita contexts (Q6592916)

Recall that a Morita context is a six-tuple \((R,S,L,N,\theta,\zeta)\) where \(R\) and \(S\) are associative rings (here not necessary unital, but with local units), \(_RN_S\) and \(_SL_R\) are (unital) bimodules, and \(\theta:N\otimes_SL\to R\) and \(\zeta:L\otimes_RN\to S\) are bimodule homomorphisms satisfying certain associativity conditions (see [\textit{P. N. Ánh} and \textit{L. Márki}, Tsukuba J. Math. 11, 1--16 (1987; Zbl 0627.16031)]). These data can be assembled into a matrix ring \(\left[\begin{array}{cc} R & N\\\NL & S \end{array}\right]\). In the paper under review it is generalized this point of view, basically by increasing the dimension of the above matrix ring. More precisely, a \textit{generalized Morita context} consists from the following data: \((A_i, M_{ij}, \varphi_{ikj})\) where:\N\begin{itemize}\N\item \(A_i\) are associative rings, for \(i=1,\ldots,n\).\N\item \(M_{ij}\) (\(i,j=1,\ldots,n\)) are \(A_i\)-\(A_j\)-bimodules, with \(M_{ii}=A_i\).\N\item \(\varphi_{ikj}:M_{ik}\otimes_{A_k}M_{kj}\to M_{ij}\) are \(A_i\)-\(A_j\)-bimodules homomorphisms (\(i,k,j=1,\ldots,n\)), where \(\varphi_{kkj}\) and \(\varphi_{ikk}\) are the left, respectively right multiplications of the elements in the modules \(M_{kj}\), respectively \(M_{ik}\) with elements in \(A_k\), and \(\varphi_{kkk}\) is the multiplication in the ring \(A_k\).\N\item Associativity condition: \(\varphi_{ihj}({\mathbf 1}\otimes\varphi_{hkj})=\varphi_{ikj}(\varphi_{ihk}\otimes{\mathbf 1})\), for all \(i,h,k,j=1,\ldots,n\).\N\end{itemize}\NThe main result of the paper under review is the generalization (by increasing the dimension of the considered matrices as above) of the following result: Let \(R\) and \(S\) be unital associative rings and let \(e=e^2\in R\) be an idempotent. If \(_{eRe}M_S\), \(_SN_{eRe}\) are bimodules defining a Morita equivalence \(eRe\sim S\) then\N\[\N\left[\begin{array}{cc} (1-e)R(1-e) & (1-e)Re\otimes M\\\NeR(1-e) & M \end{array}\right]\hbox{ and }\left[\begin{array}{cc} (1-e)R(1-e) & (1-e)Re\\\NN\otimes eR(1-e) & N \end{array}\right]\N\]\Nare bimodules defining a Morita equivalence\N\[\NR\sim\left[\begin{array}{cc} (1-e)R(1-e) & (1-e)Re\otimes M\\\NN\otimes eR(1-e) & S \end{array}\right].\N\]