Let $C$ be a semidualizing module over commutative Noetherian local ring $R$. In this paper we introduce new class of modules, namely $C$-canonical modules which are generalization canonical modules. It is shown that if the exists then and converse holds under special conditions. Also, characterization Gorenstein rings given via

Since Eilenberg and Moore [EM], the relative homological algebra, especially the Gorenstein homological algebra ([EJ2]), has been developed to an advanced level. The analogues for the basic notion, such as projective, injective, flat, and free modules, are respectively the Gorenstein projective, the Gorenstein injective, the Gorenstein flat, and the strongly Gorenstein projective modules. One c...

Let $mathcal {A}$ be an abelian category with enough projective objects and $mathcal {X}$ be a full subcategory of $mathcal {A}$. We define Gorenstein projective objects with respect to $mathcal {X}$ and $mathcal{Y}_{mathcal{X}}$, respectively, where $mathcal{Y}_{mathcal{X}}$=${ Yin Ch(mathcal {A})| Y$ is acyclic and $Z_{n}Yinmathcal{X}}$. We point out that under certain hypotheses, these two G...

In this paper, we study the relation between m-strongly Gorenstein projective (resp. injective) modules and n-strongly Gorenstein projective (resp. injective) modules whenever m 6= n, and the homological behavior of n-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Goren...

Noncommutative algebraic geometry studies a certain quotient category Rqgr of the category of graded R-modules which for commutative R is equivalent to the category of quasi-coherent sheaves by a famous theorem of Serre. For a large class of graded algebras, the so-called schematic algebras, we are able to construct a kind of scheme such that the coherent sheaves on it are equivalent to R-qgr. ...

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasi module for vertex algebras is introduced and studied. More specifically, a notion of quasi local subset(space) of Hom (W,W ((x))) for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any m...

If k is a field and R commutative k-algebra, we explore the question of when ring DR|k k-linear differential operators on isomorphic to its opposite ring. Under mild hypotheses, prove this case whenever Gorenstein local or invariants. As key step in proof show that many cases interest canonical modules admit right D-module structures.