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Matricial repetitiveness and strong \(\pi\)-regularity of the ring of a Morita context. (English) Zbl 0985.16009

A ring \(A\) is strongly \(\pi\)-regular if for any \(a\in A\) there exists an integer \(n\geq 1\) and an element \(b\in A\) such that \(a^n=a^{n+1}b\) (\(a^n=ba^{n+1}\)). A ring \(A\) is called matricially strongly \(\pi\)-regular if all the matrix rings \(M_k(A)\) for \(k\geq 1\) are strongly \(\pi\)-regular. Similarly are defined the matricially right repetitive rings.
Let \(T=\left[\begin{smallmatrix} A&N\\ M&B\end{smallmatrix}\right]\) be the ring of a Morita context and the modules \(M_A\) and \(N_B\) are finitely generated. Then: (1) if \(A\) and \(B\) are matricially strongly \(\pi\)-regular, then so is \(T\); (2) if \(A\) and \(B\) are matricially right repetitive, then so is \(T\).

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D20 Bimodules in associative algebras
16S50 Endomorphism rings; matrix rings
16D90 Module categories in associative algebras
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