Let R be a ring. An element a of R is called a left PP-element if Ra is projective. The ring R is said to be a left almost PP-ring provided that for any element a of R, either a or 1 - α is left PP. We develop, in th...Let R be a ring. An element a of R is called a left PP-element if Ra is projective. The ring R is said to be a left almost PP-ring provided that for any element a of R, either a or 1 - α is left PP. We develop, in this paper, left almost PP-rings as a generalization of von Neumann local （VNL） rings and left PP-rings. Some properties of left almost PP-rings are studied and some examples are also constructed.展开更多
Let k be a commutative ring with finite weak dimension and let G be a group. In this paper, we explore the criterion that a group G has finite Gorenstein homological dimension.It is shown that the finiteness of the Go...Let k be a commutative ring with finite weak dimension and let G be a group. In this paper, we explore the criterion that a group G has finite Gorenstein homological dimension.It is shown that the finiteness of the Gorenstein homological dimension of G coincides with the finiteness of the Gorenstein weak dimension of the group ring kG. Furthermore, we give a Gorenstein analogy of the Serre's theorem. Some well-known results for the Gorenstein homological dimension of G over the integer ring are also extended.展开更多
In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and con...In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and construct typical examples.We next describe all Σ-associated primes of the Ore extension R[x; α,δ],the skew Laurent polynomial ring R[x,x-1; α] and the skew power series ring R[[x; α]],in terms of the Σ-associated primes of R in a very straightforward way.Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.展开更多
Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under...Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under the assumption that R^σ G is a,separable extension over R^σ H,it is shown that R^σ G and R^σ H share the same homological dimensions.Several known results are then obtained as corollaries.Moreover,we investigate the relationships between the homological dimensions of Ra G and the homological dimensions of a commutative ring R,using the trivial R^σ G-module.展开更多
基金Supported by the Natural Science Foundation of Hunan Province(Grant No.2016JJ2050)
文摘Let R be a ring. An element a of R is called a left PP-element if Ra is projective. The ring R is said to be a left almost PP-ring provided that for any element a of R, either a or 1 - α is left PP. We develop, in this paper, left almost PP-rings as a generalization of von Neumann local （VNL） rings and left PP-rings. Some properties of left almost PP-rings are studied and some examples are also constructed.
基金Supported by the Natural Science Foundation of Hunan Province (Grant No. 2021JJ30536)the Scientific Research Foundation of Hunan Provincial Education Department (Grant No. 21A0493)。
文摘Let k be a commutative ring with finite weak dimension and let G be a group. In this paper, we explore the criterion that a group G has finite Gorenstein homological dimension.It is shown that the finiteness of the Gorenstein homological dimension of G coincides with the finiteness of the Gorenstein weak dimension of the group ring kG. Furthermore, we give a Gorenstein analogy of the Serre's theorem. Some well-known results for the Gorenstein homological dimension of G over the integer ring are also extended.
基金Supported by the National Natural Science Foundation of China(Grant No.11071062)the Scientific Research Fundation of Hunan Provincial Education Department(Grant No.12B101)
文摘In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and construct typical examples.We next describe all Σ-associated primes of the Ore extension R[x; α,δ],the skew Laurent polynomial ring R[x,x-1; α] and the skew power series ring R[[x; α]],in terms of the Σ-associated primes of R in a very straightforward way.Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.
基金supported by the Scientific Research Foundation of Hunan Provincial Education Department(no.18C0997).
文摘Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under the assumption that R^σ G is a,separable extension over R^σ H,it is shown that R^σ G and R^σ H share the same homological dimensions.Several known results are then obtained as corollaries.Moreover,we investigate the relationships between the homological dimensions of Ra G and the homological dimensions of a commutative ring R,using the trivial R^σ G-module.
