All rings considered are associative with identity. All modules are left unital. An \(R\)-module \(M\) is said to be NFG if it is not finitely generated. The author presents for a particular ring \(R\) three new examples of unusual NFG projective \(R\)-modules: Example 1. There exist projective \(R\)-modules \(P\), \(Q\) and \(P\cap Q\) such that \(P \times Q \cong (P\cap Q) \times R\) where \(P\), \(Q\) and \(P \cap Q\) are NFG, indecomposable, and pairwise non-isomorphic; Example 2. There exists a nonzero projective \(R\)-module \(M\) such that each nonzero direct summand of \(M\) is NFG and decomposable; Example 3. There exists a projective \(R\)- module \(M\) such that each nonzero direct summand of \(M\) is NFG but, if \(M = A \oplus B\), one summand is a finite direct sum of indecomposable submodules and the other summand is decomposable.
For the entire collection see [Zbl 0772.00017].