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Weak-projective dimensions

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In this paper, the notions of weak-projective modules and weak-projective dimension over commutative domain R are given. It is shown that over semisimple rings with weak global dimension 1, these modules are equivalent to weak-injective modules. The weak-projective dimension measures how far away a domain is from being a Prufer domain. | Turk J Math 35 (2011) , 627 – 636. ¨ ITAK ˙ c TUB doi:10.3906/mat-0805-34 Weak-projective dimensions Mohammad Javad Nikmehr, Zahra Poormahmood and Reza Nikandish Abstract In this paper, the notions of weak-projective modules and weak-projective dimension over commutative domain R are given. It is shown that over semisimple rings with weak global dimension 1, these modules are equivalent to weak-injective modules. The weak-projective dimension measures how far away a domain is from being a Pr¨ ufer domain. Several properties of these modules are also presented. Key Words: Semi-Dedekind domain; Weak-injective modules; Weak- projective dimension, projective modules; Pr¨ ufer domain 1. Introduction In this note, R will denote a commutative domain with identity and Q ( = R) will denote its field of quotients. The R -module Q/R will be denoted by K . Lee in [5] studied the structure of weak-injective modules. An R -module M is called weak-injective if Ext1R (N, M ) = 0 for all R -modules N of weak dimension ≤ 1 . In section 2, we introduce a class of R -modules under the name of weak-projective R -modules. We show that weak-projective R -modules are identical to projective R -modules if and only if R is semisimple. Recall that R is called Pr¨ ufer domain if every finitely generated ideal of R is projective. There are numerous characterizations of Pr¨ ufer domains, which can be found in [3]. We show that each weak-projective R -module is F P -projective when R is a Noetherian ring. The domain R is called semi-Dedekind if every h-divisible R -module is pure-injective. For more details of these domains, we refer the reader to [4]. In section 3, we introduce the concept weak-projective dimension wpd(M ) of an R -module M and give some results. We show that this dimension has the properties that we expect of a “dimension” when the domain is semi-Dedekind. Throughout this paper, M is an R -module. The notation (w.)D(R) stands for the (weak) global dimension of R . Also,