Noncommutative G-hereditary rings

In this paper, we introduce and study left (right) [Formula: see text]-hereditary rings over any associative ring, and these rings are exactly [Formula: see text]-hereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. As applications, we give a partial answer to the question posed by Bazzoni, Cortés-Izurdiaga and Estrada, and characterize when the character module of a Gorenstein injective left [Formula: see text]-module is Gorenstein flat provided that [Formula: see text] is a left [Formula: see text]-hereditary ring. In addition, some new characterizations of left hereditary rings are given.

Σ-Rickart modules

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a [Formula: see text]-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module [Formula: see text] is called [Formula: see text]-Rickart if every direct sum of copies of [Formula: see text] is Rickart. It is shown that any direct summand and any direct sum of copies of a [Formula: see text]-Rickart module are [Formula: see text]-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring [Formula: see text] is right hereditary if and only if every submodule of any projective right [Formula: see text]-module is projective if and only if every factor module of any injective right [Formula: see text]-module is injective. Also, we have a characterization of a finitely generated [Formula: see text]-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

Gorenstein homological theory for differential modules

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.

ALL GORENSTEIN HEREDITARY RINGS ARE COHERENT

A ring R is called Gorenstein hereditary (G-hereditary) if every submodule of a projective module is Gorenstein projective (i.e. Ggldim (R) ≤ 1). In this paper, we settle a question raised by Mahdou and Tamekkante in [On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sci. Eng.36 (2011) 436] about the coherence of G-hereditary rings. It is shown that a ring R is Gorenstein semihereditary if and only if every finitely generated submodule of a projective module is Gorenstein projective. As a consequence of this result, we have that every G-hereditary ring R is coherent.

Right hereditary affine PI rings are left hereditary

Small [11] gave the first example of a right hereditary PI ring which is not left hereditary. Robson and Small [9] proved that a prime PI right hereditary ring is a classical order over a Dedekind domain, and hence is Noetherian (and therefore left hereditary). The authors have shown [4] that a right hereditary semiprime PI ring which is finitely generated over its center is left hereditary. In this paper we consider right hereditary PI rings T which are affine (i.e. finitely generated as an algebra over a central subfield k).

Extended semi-hereditary rings

A ring R for which every finitely generated right submodule of SR, the left flat epimorphic hull of R, is projective is termed an extended semi-hereditary ring. It is shown that several of the characterizing properties of Prufer domains may be generalized to give characterizations of extended semi-hereditary rings. A suitable class of PP rings is introduced which in this case serves as a generalization of commutative integral domains.