The authors prove that for a commutative ring \(R\), every maximal ideal of \(R\) is finitely generated and projective if and only if \(R\) has projective socle and neatness and \(P\)-purity coincide. For a commutative ring \(R\), they prove that neatness and \(P\)-purity coincide if and only if every maximal ideal of \(R\) is finitely generated and the unique maximal ideal \(P_P\) of the local ring \(R_P\) is a principal ideal for every maximal ideal \(P\) of \(R\).
There are four sections in this article. In section \(3\), the authors prove in Theorem \(3.10\) that if \(R\) is a commutative ring such that every maximal ideal of R is finitely generated and projective, then an Auslander-Bridger transpose \(Tr(s)\) is projectively equivalent to \(S\) for every simple \(R\)-module \(S\) and \(_R \mathcal{N}eat = _R \mathcal{P}Pure.\)
For proving this result, they establish many results in section \(3\) as follows: for a commutative local ring \(R\) with a unique maximal ideal \(m\), they show in Corollary \(3.5\) that \(_R \mathcal{N}eat = _R \mathcal{P}Pure,\) if and only if \(m\) is a principal ideal. If the unique maximal ideal \(m\) of a commutative local ring \(R\) is projective, then it is necessarily a principal ideal. Over any commutative ring, in Lemma \(3.6\), they analyse how neatness and \(P\)-purity behave under localization and obtain the main result in Theorem \(3.7:\) for a commutative ring \(R\), \(_R \mathcal{N}eat = _R \mathcal{P}Pure,\) if and only if every maximal ideal \(m\) of \(R\) is finitely generated and the maximal ideal \(m\) of the local ring \(R_m\) is a principal ideal. In the last Section 4, they prove that all maximal ideals of a commutative ring \(R\) are finitely generated and projective if and only if \(R\) has projective socle and \(_R \mathcal{N}eat = _R \mathcal{P}Pure.\)
In general, this is an interesting aticle which paves way to many possible research in this field which makes this article worth reading.