On \(\phi\)-(weak) global dimension (Q6607144)

Let \(R\) be a commutative ring with identity and with prime nilradical. In the paper under review, the authors introduced and studied the homological dimensions defined in the context of commutative rings with prime nilradical. Let \(\mathcal{H}\) be the set of all rings \(R\) of Nil\((R)\) is divided prime ideal and call every ring \(R\) which belongs to \(\mathcal{H}\) is a \(\phi\)-ring. Using these dimensions, they characterized several \(\phi\)-rings such us \(\phi\)-Prüfer, \(\phi\)-chained, \(\phi\)-von Neumann rings, etc. They also studied the \(\phi\)-(weak) global dimension of the trivial ring extensions defined by some conditions. Among others. they proved that \(R\) is \(\phi\)-semisimple if and only if \(\phi\)-\(gl. \dim(R) = 0\); and for \(R\in\mathcal{H}\), \(R\) is a \(\phi\)-von Neumann regular ring if and only if \((R, \mathrm{Nil}(R))\) is a local ring if and only Nil\((R)\) is the unique prime ideal of \(R\). Also they proved that the trivial ring \(R\ltimes M\) of \(R\) by a module \(M\) is a \(\phi\)-von Neumann regular ring, if and only if \(R\) is a \(\phi\)-von Neumann regular ring. The paper contains tens of definitions, theorems and generalizations/analogues of classical homological results, and it is divided in 5 sections and preceded by an introduction.