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Volume :13 Issue : 1 1986      Add To Cart                                                                    Download

A note on FBN-rings and rings of Krull dimension one

Auther : N. S. KHATIB AND SURJEET SINGH

Department of Mathematics, University of Kuwait, P.O. Box 5969, Kuwait

 

ABSTRACT

 

Let R be any ring and for any module MR, let X(M) denote the hereditary torsion theory on Mod-R cogenerated by M. Consider the following conditions: (I) For any two uniform modules UR, VR that are X(U)-cocritical and X(V)-cocritical respectively, X(V) < X(U) whenever HomR(U, V) # 0. (11) For any two uniform modules UR, VR, z(V) < z(U) whenever HomR(U, V) # 0. (111) Let U, be a uniform. X(U -injective module and let VR, be any module which is X(V)-cocritical; if for some non-zero submodule W of V, there exists a uniform extension of U by W, then there exists a uniform extension of U by V. All these conditions are satisfied by any commutative noetherian ring. In an earlier paper by the present authors (Khatib & Singh 1983). some structural similarit~esb etween noetherian rings satisfying some of these conditions, and commutative noetherian rings have been noticed. In the present note it is shown that conditions (I) and (11) are equivalent for (FBN)-rings and prime right noetherian rings of Krull dimension one. Some necessary and sufficient conditions for these rings to satisfy (I) or (11) are given. If any of these rings satisfies (I), then it is shown to satisfy (111).

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