Minimal generating sets for free modules

Abstract:

Let $R$ be a ring admitting a free module with generating set shorter than the length of a basis. If $n$ is the shortest basis among all such modules and $m$ the length of its shortest generating set then $n = m + 1$ and every free module with basis of length $\geqq m + 1$ has a generating set of length $m$. If $R$ has module type $(h,k)$ then $m = h$, that is an $R$-module with basis of length $u < h$ not only has all bases of length $u$ but also has no generating set of length $< u$. The integer $m$ together with the module type define a new ring invariant which satisfies many of the properties of the module type.