Quasi-projective modules over prime hereditary noetherian V-rings are projective or injective

Some results on V-rings and weakly V-rings

“…However, we were able to show that if, in addition, the ring R is a right V-ring, then a right R-module P is projective if and only if P is R-projective (see our recent work [8]). Hence the V-condition makes prime right hereditary right Noetherian rings behave so nicely that they become ''test-rings'' for the projectivity in the category of their modules (just like all rings do for injectivity on their modules).…”

Some results on V-rings and weakly V-rings

“…However, we were able to show that if, in addition, the ring R is a right V-ring, then a right R-module P is projective if and only if P is R-projective (see our recent work [8]). Hence the V-condition makes prime right hereditary right Noetherian rings behave so nicely that they become ''test-rings'' for the projectivity in the category of their modules (just like all rings do for injectivity on their modules).…”

max-projective modules

“…In [11,Lemma 1], the authors prove that over a right nonsingular right V -ring, maxprojective right R-modules are nonsingular. Regarding the converse of this fact, we have the following.…”

Rings whose nonsingular right modules are $R$-projective

“…In [14], C. Faith asked when R-projectivity implies projectivity for all right R-modules. This problem has been considered by several authors, see [25], [18], [21], [8], [2], [31], [32]. DOI 10.14712/1213DOI 10.14712/ -7243.2021 Characterizing rings by projectivity of some classes of their modules is a classical problem in ring and module theory.…”

Some results on V-rings and weakly V-rings

“…However, we were able to show that if, in addition, the ring R is a right V-ring, then a right R-module P is projective if and only if P is R-projective (see our recent work [8]). Hence the V-condition makes prime right hereditary right Noetherian rings behave so nicely that they become ''test-rings'' for the projectivity in the category of their modules (just like all rings do for injectivity on their modules).…”

max-projective modules

“…In [11,Lemma 1], the authors prove that over a right nonsingular right V -ring, maxprojective right R-modules are nonsingular. Regarding the converse of this fact, we have the following.…”

Rings whose nonsingular right modules are $R$-projective

“…In [14], C. Faith asked when R-projectivity implies projectivity for all right R-modules. This problem has been considered by several authors, see [25], [18], [21], [8], [2], [31], [32]. DOI 10.14712/1213DOI 10.14712/ -7243.2021 Characterizing rings by projectivity of some classes of their modules is a classical problem in ring and module theory.…”

Some results on V-rings and weakly V-rings

“…However, we were able to show that if, in addition, the ring R is a right V-ring, then a right R-module P is projective if and only if P is R-projective (see our recent work [8]). Hence the V-condition makes prime right hereditary right Noetherian rings behave so nicely that they become ''test-rings'' for the projectivity in the category of their modules (just like all rings do for injectivity on their modules).…”

max-projective modules

“…In [11,Lemma 1], the authors prove that over a right nonsingular right V -ring, maxprojective right R-modules are nonsingular. Regarding the converse of this fact, we have the following.…”

Rings whose nonsingular right modules are $R$-projective

“…In [14], C. Faith asked when R-projectivity implies projectivity for all right R-modules. This problem has been considered by several authors, see [25], [18], [21], [8], [2], [31], [32]. DOI 10.14712/1213DOI 10.14712/ -7243.2021 Characterizing rings by projectivity of some classes of their modules is a classical problem in ring and module theory.…”