In this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. Several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given. Then we investigate for a ring $R$, when any fi...

The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian groups, eliminating unnecessary additional hypotheses used in earlier versions.

It is shown that a squarefree principal Borel ideal satisfies the persistence property for the associated prime ideals. For the graded maximal ideal we compute the index of stability with respect to squarefree principal Borel ideals and determine their stable set of associated prime ideals.

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

We consider a commutative ring R graded by an arbitrary abelian group G, and define the grade of a G-homogeneous ideal I on R in terms of vanishing of C̆ech cohomology. By defining the dimension in terms of chains of homogeneous prime ideals and supposing R satisfies a.c.c. on G-homogeneous ideals and has a unique G-homogeneous maximal ideal, we can define graded versions of the depth of R and C...

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K -group of semi-infinite flag manifolds, which is described explicitly terms Lakshmibai-Seshadri paths (or equivalently, quantum paths); contrast to dominant our previous paper [17] , has significant finiteness property. Based on geometric results established proof representation-theoretic, and follows from certain...