On rigid rings (Q1336807)

A ring \(R\) is said to be rigid if whenever for a submodule \(K\) of a free \(R\)-module \(F\) of rank \(n\), \(F/K\) is generated by \(m<n\) elements, \(K\) contains a unimodular element of \(R\). This concept originated in the study of pole-assignment of linear systems over rings, but the author here shows that it is also interesting algebraically. Several classes of rings are shown to be rigid, for example BCU-rings, homomorphic images of rigid rings, power series rings of rigid rings, products of rigid rings, semi-local rings, PIDs etc. Sufficient conditions for \(R[X]\) to be rigid are also given. The paper is concluded by several examples of non-rigid rings.