Forcing linearity numbers of semisimple modules over integral domains. (Q1411436)

Let \(R\) be a ring with identity, \(V\) a unitary \(R\)-module. Consider \(M_R(V)\), the near-ring of all functions from \(V\) to \(V\) which commute with the action of \(R\). A set \(S\) of submodules of \(V\) is said to force linearity if whenever \(f\in M_R(V)\) is linear on all submodules of \(S\), then \(f\in\text{End}_R(V)\). If \(M_R (V)=\text{End}_R(V)\) then the forcing linearity number of \(V\) (\(\text{fln}(V)\)) is 0, otherwise it is the size of a minimal set \(S\) of submodules which force linearity [\textit{P. R. Fuchs} and \textit{C. J. Maxson}, J. Pure Appl. Algebra 141, No. 3, 211-224 (1999; Zbl 0953.16003), \textit{C. J. Maxson} and \textit{J. H. Meyer}, J. Algebra 223, No. 1, 190-207 (2000; Zbl 0953.16034)]. Forcing linearity numbers have been determined in a number of cases. In this paper, they are determined completely for semisimple modules over an integral domain. The possible values are shown to be \(0,1,2,q+2\), where \(q\) is the order of a simple \(R\)-module, and \(\infty\).