Self-injective von Neumann regular subrings and a theorem of Pere Menal (Q1802375)

The author studies von Neumann regular (VNR) subrings of a right self- injective (SI) ring. The paper consists of seven sections with the last two sections being devoted to open problems and tribute to Professor Pere Menal whose theorem is the starting point of this paper. The main results of this paper are the following ones. (i) A tensor product \(A\otimes_ k B\) of modules over a commutative ring \(k\) generates mod-\(k\) iff both \(A\) and \(B\) generate mod-\(k\). As a consequence of this, Menal's general tensor theorem is deduced which states that if \(K\) is a right SI split- flat algebra over a commutative ring \(k\), and if \(K = A\otimes_ kB\), for subalgebras \(A\), \(B\), then \(A\), \(B\) are right SI split-flat algebras. (ii) If \(K\) is a right SI, VNR, and a centralizing extension of a genuine max VNR subring \(A\), then either \(A\) is right SI or \(K\) is maximal right quotient ring of \(A\). (iii) If \(K\) is an Abelian SI ring, then a max VNR subring \(A\) of \(K\) is continuous. (iv) If \(A\) is a maximal (commutative) VNR subring of a non-VNR ring \(K\), then \(A\) contains all central idempotents, and moreover all idempotents of \(K\) that centralize \(A\). (v) If a right SI Abelian VNR ring \(K\) is right singular over a max VNR subring \(A\), then \(A\) is right SI and isomorphic to a ring direct factor \(K_ 2\) of \(K\): \(K = K_ 1\times K_ 2 \approx K_ 1 \times A\), where \(K_ 1\), the singular right \(A\)-submodule of \(K\) is a skew field.