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

ABSTRACT

Let R be any ring and for any module M_R, let _X(M) denote the hereditary torsion theory on Mod-R cogenerated by M. Consider the following conditions: (I) For any two uniform modules U_R, V_R that are _X(U)-cocritical and _X(V)-cocritical respectively, _X(V) < _X(U) whenever Hom_R(U, V) # 0. (11) For any two uniform modules UR, VR, z(V) < z(U) whenever Hom_R(U, V) # 0. (111) Let U, be a uniform. _X(U -injective module and let V_R, 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).