Some Properties Of Coherent And Pseudo-coherent Rings

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).