On minimal spanning systems over semiperfect rings (Q1177322)

Let \(R\) be a semiperfect ring. Then each finitely generated right \(R\)-module is isomorphic to a module of the form \(\bigoplus^n_{i=1}e_iR\) with local idempotents \(e_1,\ldots,e_n\) of \(R\). If \(f_1,\ldots,f_m\) is another family of local idempotents of \(R\), then any homomorphism \(\bigoplus^m_{j=1}f_j R\to\bigoplus^n_{i=1} e_iR\) is determined by a matrix \((e_ir_{ij}f_j)_{1\leq i\leq n,\ 1\leq j\leq m}\) with \(r_{ij}\in R\). In the present paper a calculus for such matrices together with some applications is developed. For instance the problem, whether a finitely presented module is indecomposable resp. has a non-zero projective summand, can be translated into matrix language. In addition it is proved that \(R\) has only finitely many two-sided ideals, provided that there exist only finitely many isomorphism classes of right \(R\)-modules with simple top. This generalizes a result by \textit{J. P. Jans} [Ann. Math. (2) 66, 418--429 (1957; Zbl 0079.05203)].