When every finitely generated flat module is projective. (Q1882877)

The paper is devoted to those rings over which every `finitely generated' flat module is projective: the authors call these \(S\)-rings (after Sakhajev, who investigated them). Say that a sequence \(a_1,a_2,\dots\) of elements of a ring \(R\) is a right \(a\)-sequence if \(a_{i+1}a_i=a_i\) for all \(i\) and say this converges if the corresponding increasing sequence of right ideals \(a_iR\) is eventually constant. It was shown by \textit{A. Facchini, D. Herbera} and \textit{I. Sakhajev} [in Commun. Algebra 31, No. 9, 4195-4214 (2003; Zbl 1032.16003)] that \(R\) is an \(S\)-ring iff for every \(n\) each \(a\)-sequence in the ring, \(M_n(R)\), of \(n\times n\) matrices over \(R\) converges. In this paper a variety of results and examples are given. For instance, every ring with right Krull dimension is a right and left \(S\)-ring, every commutative uniform ring is an \(S\)-ring, the free associative algebra \(k\langle X\rangle\) over a field \(k\) in any set \(X\) of indeterminates is an \(S\)-ring.