Forcing linearity numbers of semicyclic modules over commutative Noetherian rings
Given a module \(V\) over a commutative ring \(R\), a function \(f:V\rightarrow V\) is said to be ``homogeneous'' if \(f(rv) = rf(v)\) for all \(r\in R\) and all \(v\in V\) and \(M_R(V)\) denotes the set of all such functions. Clearly \(M_R(V)\) is contained in \(\text{End}_R(V)\), the set of module endomorphisms on \(V\). This paper looks at how close these two sets are in the case where \(R\) is Noetherian and \(V\) is semicyclic, meaning that \(V\) is decomposable as a direct sum of cyclic submodules. This closeness is measured by the ``forcing linearity number'' as defined by \textit{C. J. Maxson} and \textit{J. H. Meyer} [J. Algebra 223, 190--207 (2000; Zbl 0953.16034)], based on the minimum number of submodules of \(V\) on which there is a homogeneous \(f\) which is not linear. The author here extends some results for finitely generated modules obtained by \textit{C. J. Maxson} and \textit{A. B. Van der Merwe} [Rocky Mt. J. Math. 35, 929--939 (2005; Zbl 1098.13020)] to the non-finitely generated case, as well as other recent results by Maxson and Meyer.
scientific article; zbMATH DE number 2063944