Finite generation of projective modules over certain rings (Q1382684)

Let \(R\) be an associative ring with \(1\), \(R_n\) the ring of \(n\times n\) matrices over \(R\) and \(A_m\) (\(m=1,2,\ldots\)) a sequence of elements of \(R_n\) satisfying \(A_m=A_{m+1}\cdot A_m\) (\(m=1,2,\ldots\)). If \(F=\langle x_1,\ldots,x_n\rangle\) is a free right \(R\)-module, the elements \(u_{m1},\ldots,u_{mn}\) of \(F\) are defined by the equality \((u_{m1},\ldots,u_{mn})=(x_1,\ldots,x_n)\cdot A_m\) (\(m=1,2,\ldots\)). Set \(P(A_m)=\langle u_{m1},\ldots,u_{mn}\rangle\). Then the submodule \(\sum_{i=1}^\infty P(A_m)\) of \(F\) is denoted by \(P(A_m=A_{m+1}\cdot A_m)\). These submodules are shown to be projective. This paper studies the question of finite generation of projective modules of the type \(P(A_m=A_{m+1}\cdot A_m)\) as a continuation and generalization of an earlier paper by the same author [Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 8(375), 65-75 (1993; Zbl 0834.16004)]. Various equivalences to \(P(A_m=A_{m+1}\cdot A_m)\) being finitely generated are stated and the very long and complicated proofs are shown. Most of the results require \(R\) to be a PI-ring.