Projective modules with unique maximal submodules are cyclic

Abstract:

There is a question formulated by R. Ware in 1971. That is, if a projective right $R$-module $P$ has a unique maximal submodule $L$, then $L$ is the largest maximal submodule of $P$. First we give a structure theorem for such a projective module. Next we define the notion of a minimal generator set and give an affirmative answer to this question by using this notion. In the last section, we give an example of a non-quasi-small module to discuss whether the answer to this question is affirmative or negative. The key theorem in Comm. Algebra 31 (2003), pp. 4195–4241 asserts that answer to this question is negative. But our example seems to be a counterexample of this key theorem.