Forcing linearity numbers for modules over simple domains. (Q1411434)

Let \(R\) be a ring with identity, and let \(V\) be a unital left \(R\)-module. A function \(f\colon V\to V\) is homogeneous if \(f(rv)=rf(v)\) for all \(r\in R\), \(v\in V\). The forcing linearity number \(\text{fln}(V)\) of \(V\) is a measure of how much local linearity is needed for each homogeneous \(f\) to be linear [cf. \textit{P. R. Fuchs}, Proc. Am. Math. Soc. 128, No. 1, 5-15 (2000; Zbl 0968.16001)]. If \(R\) is a simple ring but not a division ring, then \(\text{fln}(V)=0\) for any semisimple \(R\)-module \(V\). If \(V\) is nonsingular and not semisimple, then \(\text{fln}(V)\in\{0,1\}\). If \(R\) is a domain and \(V\) has a nontrivial torsionfree direct summand, then \(\text{fln}(V)\geq 1\). Finally \(\text{fln}(V)\) is studied for modules \(V\) over a simple hereditary Noetherian domain, and some examples are given to illustrate the theory.