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Some Properties of Coherent and Pseudo-coherent Rings

Author: ZhengCuiLing
Tutor: DingNanQing
School: Nanjing University
Course: Basic mathematics
Keywords: coherent ring J-coherent ring pesudo-coherent ring singly in-jective module singly flat module singly injective dimension singly flat dimension
CLC: O153.3
Type: Master's thesis
Year: 2011
Downloads: 22
Quote: 0
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Abstract


Let R be a ring, R is called a left coherent ring if every finitely generated left ideal is finitely presented. If J(R), the Jacobson radical of R, is a coherent left R-module, ie, every finitely generated submodule of J(R) is finitely presented, then R is said to be a left J-coherent ring. In this paper, we first recall some definitions and propositions of coherent rings, then introduce the notion of pesudo-coherent rings. A ring R is said to be left pesudo-coherent if the left annihilator of any finite subset of R is a finitely generated left ideal. Singly injective module and singly flat module are introduced to investigate pesudo-coherent rings. Necessary and sufficient conditions for a ring R to be left pesudo-coherent are given. It is shown that there are many similarities between coherent and pesudo-coherent rings. Singly left injective dimension and singly right flat dimension are also studied. Finally, we define the left singly injective dimension l.SID(R) and right singly flat dimension r.SFD(R) of a ring R. It is shown that, if the class of all singly left injective modules is closed under cokernels of monomor-phisms, then l.SID(R)= r.SFD(R).

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Abstract algebra ( Algebra ) > Ring Theory
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