Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator annMn(S)(X)annMn(S)(X) is a finitely g...Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator annMn(S)(X)annMn(S)(X) is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of M^n if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.展开更多
Let M be a right R-module and N an infinite cardinal number. A right R-module N is called N-M-coherent if for any 0 ≤ A < B ≤ N, such that B/A → mR for some m ∈ M, if B/A is finitely generated, then B/A is N-fp...Let M be a right R-module and N an infinite cardinal number. A right R-module N is called N-M-coherent if for any 0 ≤ A < B ≤ N, such that B/A → mR for some m ∈ M, if B/A is finitely generated, then B/A is N-fp. A ring R is called N-M-coherent if RR is N-M-coherent. It is proved under some additional conditions that the N-product of any family of M-flat left R-modules is M-flat if and only if R is N-M-coherent. We also give some characterizations of N-M-coherent modules and rings.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 this paper we study the existence of FIn-envelopes, FI1/n-envelopes and FIn-covers, where FIn denotes the class of all n-absolute pure modules for an integer n 〉 0 or n = ∞. We prove that FI1/n-envelopes and FIn-...In this paper we study the existence of FIn-envelopes, FI1/n-envelopes and FIn-covers, where FIn denotes the class of all n-absolute pure modules for an integer n 〉 0 or n = ∞. We prove that FI1/n-envelopes and FIn-covers exist over an n-coherent ring R, and FIn-covers and special FIn-preenvelopes exist over any ring R.展开更多
文摘Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator annMn(S)(X)annMn(S)(X) is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of M^n if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.
基金the National Natural Science Foundation of China (No.10171082)
文摘Let M be a right R-module and N an infinite cardinal number. A right R-module N is called N-M-coherent if for any 0 ≤ A < B ≤ N, such that B/A → mR for some m ∈ M, if B/A is finitely generated, then B/A is N-fp. A ring R is called N-M-coherent if RR is N-M-coherent. It is proved under some additional conditions that the N-product of any family of M-flat left R-modules is M-flat if and only if R is N-M-coherent. We also give some characterizations of N-M-coherent modules and rings.
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 this paper we study the existence of FIn-envelopes, FI1/n-envelopes and FIn-covers, where FIn denotes the class of all n-absolute pure modules for an integer n 〉 0 or n = ∞. We prove that FI1/n-envelopes and FIn-covers exist over an n-coherent ring R, and FIn-covers and special FIn-preenvelopes exist over any ring R.
