Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and on...Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.展开更多
In this article, we introduce the discrete subgroup in ℝ<sup>n</sup> as preliminaries first. Then we provide some theories of cyclic lattices and ideal lattices. By regarding the cyclic lattices...In this article, we introduce the discrete subgroup in ℝ<sup>n</sup> as preliminaries first. Then we provide some theories of cyclic lattices and ideal lattices. By regarding the cyclic lattices and ideal lattices as the correspondences of finitely generated R-modules, we prove our main theorem, i.e. the correspondence between cyclic lattices in ℝ<sup>n</sup> and finitely generated R-modules is one-to-one. Finally, we give an explicit and countable upper bound for the smoothing parameter of cyclic lattices.展开更多
Let R be a ring, n, d be fixed non-negative integers, Jn,d the class of （n, d）- injective left R-modules, and Fn,d the class of （n, d）-flat right R-modules. In this paper, we prove that if R is a left n-coherent r...Let R be a ring, n, d be fixed non-negative integers, Jn,d the class of （n, d）- injective left R-modules, and Fn,d the class of （n, d）-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then gl-right-Jn,a-dimRM ≤ m if and only if gl-left-Jn,d-dimRM ≤ m -- 2, if and only if Extm＋k（M, N） = 0 for all left R-modules M, N and all k 〉 -1, if and only if Extm-l（M, N） = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then - - is right balanced on MR ×RM by Fn,d × Jn,d, and investigate the global right Jn,d-dimension of RM and the global right Fn,d-dimension of MR by right derived functors of - -. Some known results are obtained as corollaries.展开更多
In the 19th problem of [1], J. S. Golan put forward the following question: 'For what kind of ring R is it true that the map γ_#:R-tors→S-tors is surjective for every ring surjective homomorphism r: R→S?' w...In the 19th problem of [1], J. S. Golan put forward the following question: 'For what kind of ring R is it true that the map γ_#:R-tors→S-tors is surjective for every ring surjective homomorphism r: R→S?' where both R and S are associative rings with identity, R-tors stands for a lattice composed of all hereditary torsion theories in left R-module category R-Mod and for each τ ∈ R-tors, γ_#(τ)=σ=(f_σ,f_σ)∈S-tors, in展开更多
In this paper,the authors define the homology of sets,which comes from and contains the ideas of path homology and embedded homology.Moreover,A Kunneth formula for sets associated to the homology of sets is given.
文摘Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.
文摘In this article, we introduce the discrete subgroup in ℝ<sup>n</sup> as preliminaries first. Then we provide some theories of cyclic lattices and ideal lattices. By regarding the cyclic lattices and ideal lattices as the correspondences of finitely generated R-modules, we prove our main theorem, i.e. the correspondence between cyclic lattices in ℝ<sup>n</sup> and finitely generated R-modules is one-to-one. Finally, we give an explicit and countable upper bound for the smoothing parameter of cyclic lattices.
文摘Let R be a ring, n, d be fixed non-negative integers, Jn,d the class of （n, d）- injective left R-modules, and Fn,d the class of （n, d）-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then gl-right-Jn,a-dimRM ≤ m if and only if gl-left-Jn,d-dimRM ≤ m -- 2, if and only if Extm＋k（M, N） = 0 for all left R-modules M, N and all k 〉 -1, if and only if Extm-l（M, N） = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then - - is right balanced on MR ×RM by Fn,d × Jn,d, and investigate the global right Jn,d-dimension of RM and the global right Fn,d-dimension of MR by right derived functors of - -. Some known results are obtained as corollaries.
文摘In the 19th problem of [1], J. S. Golan put forward the following question: 'For what kind of ring R is it true that the map γ_#:R-tors→S-tors is surjective for every ring surjective homomorphism r: R→S?' where both R and S are associative rings with identity, R-tors stands for a lattice composed of all hereditary torsion theories in left R-module category R-Mod and for each τ ∈ R-tors, γ_#(τ)=σ=(f_σ,f_σ)∈S-tors, in
基金supported by the National Natural Science Foundation of China(No.12001310)Science and Technology Project of Hebei Education Department(No.QN2019333)+2 种基金the Natural Fund of Cangzhou Science and Technology Bureau(No.197000002)a Project of Cangzhou Normal University(No.xnjjl1902)China Postdoctoral Science Foundation(No.2020M680494)。
文摘In this paper,the authors define the homology of sets,which comes from and contains the ideas of path homology and embedded homology.Moreover,A Kunneth formula for sets associated to the homology of sets is given.
