关于所有强平坦右S-系是正则系的幺半群(英).pdfVIP

Æ26Æ4 Vol.26, No.4 200611 JOURNAL OF MATHEMATICAL RESEARCH AND EXPOSITION Nov., 2006 Article ID: 1000-341X(2006)04-0720-05 Document code: A On Monoids over which All Strongly Flat Right S-Acts Are Regular QIAO Hu-sheng (College of Math. Info. Sci., Northwest Normal University, Lanzhou 730070, China ) (E-mail: qiaohs@nwnu.edu.cn) Abstract: This paper investigates the characterizations of monoids over which all strongly flat right S-acts are regular. It is shown that all strongly flat right S-acts are regular if and only if S is a right PSF monoid and every left collapsible submonoid of S contains a left zero. This result gives a new answer to the problem in Kilp and Knauer[7]. Key words: strong flat S-acts; regular S-acts. MSC(2000): 20M30, 18G05 CLC number: O152.7 1. Introduction Let S be a monoid. The definitions of right S-act A and left S-act B can be found in [1]. Strong flatness and regular property of right acts over a monoid S have been the topic of many papers on homological classification of monoids during last decades (see for example [2] and the references cited there). In [3], Stenstr¨om called a right S-act A strongly flat, if the functor A⊗- (from the category of left S-acts to the category of sets) preserves pullbacks and equalizers. In fact the author has shown that A is strongly flat if and only if it satisfies the following condition (P) and (E): ′ ′ ′ ′ ′′ ′′ ′ ′′ ′ (∀a, a ∈ A)(∀s, s ∈ S)(as = a s ⇒ (∃a ∈ A)(∃u, v ∈ S)(a = a