Search

In this paper, we introduce and study V -Gorenstein at modules and show the stability of the category of V -Gorenstein at modules. We investigate the existence of V -Gorenstein at covers and V -Gorenstein at preenvelopes for any left R -module. Also we prove that ( V - GF ;V - GF ? ) is a perfect hereditary cotorsion pair in B l ( R) , where V - GF stands the class of V -Gorenstein at left R -modules and B l ( R) is the left Bass class. Some applications are given.

In this paper, we introduce and study Ext-strongly Gorenstein projective modules. We prove that the class of Ext-strongly Gorenstein projective modules is projective resolving. Moreover, we consider Ext-strongly Gorenstein projective precovers.

In this paper we call a ring R delta(r)-clean if every element is the sum of an idempotent and an element in delta(R-R) where delta(R-R) is the intersection of all essential maximal right ideals of R. If this representation is unique (and the elements commute) for every element we call the ring uniquely (strongly) delta(r)-clean. Various basic characterizations and properties of these rings are proved, and many extensions are investigated and many examples are given. In particular, we see that the class of delta(r)-clean rings lies between the class of uniquely clean rings and the class of exchange rings, and the class of uniquely strongly delta(r)-clean rings is a subclass of the class of uniquely strongly clean rings. We prove that R is delta(r)-clean if and only if R/delta(r)(R-R) is Boolean and R/Soc(R-R) is clean where Soc(R-R) is the right socle of R.

A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner's result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

In this paper, we discuss the Bass class and the Auslander class with respect to a semidualizing module over an associative ring. Let SCR be a semidualizing module, we proved that the Bass class BC (R) is a right orthogonal subcategory of some right R-module; and that the Auslander class AC (S) is a left orthogonal subcategory of the character module of some left S-module. As an application, we introduce the notion of the minimal semidualizing module, and get a one to one correspondence between the isomorphism classes of minimal semidualizing R-modules and maximal classes among coresolving preenvelope classes of Mod R with the same Ext-projective generators in gen* R.