tailieunhanh - On the uniqueness of strongly flat covers of cyclic acts

In strongly flat covers of cyclic acts are discussed and it is asked if strongly flat covers are unique. From this point of view, in this paper we give numerous classes of monoids over which strongly flat covers of cyclic acts are unique. | Turk J Math 35 (2011) , 437 – 442. ¨ ITAK ˙ c TUB doi: On the uniqueness of strongly flat covers of cyclic acts Majid Ershad, Roghaieh Khosravi Abstract In [1], strongly flat covers of cyclic acts are discussed and it is asked if strongly flat covers are unique. From this point of view, in this paper we give numerous classes of monoids over which strongly flat covers of cyclic acts are unique. Key Words: Strongly flat, cover, cyclic act 1. Introduction A right S -act BS is called a cover of a right S -act AS if there exists an epimorphism f : BS −→ AS such that for any proper subact CS of BS the restriction f|CS : CS −→ AS is not an epimorphism. An epimorphism with this property is called a coessential epimorphism. A cover BS of an act AS is called a projective (strongly flat) cover of AS if BS is a projective (strongly flat) act. In [1], Mahmoudi and Renshaw investigate strongly flat (condition (P)) covers of cyclic acts. Recently in [2], monoids over which all right S -acts have strongly flat (condition (P)) covers are characterized, answering a question of Mahmoudi and Renshaw. Another question posed in [1] is whether strongly flat covers of acts are unique. This question remains open up to now. In Section 2 we give some conditions on a monoid under which strongly flat covers of its cyclic acts are unique. The reader is referred to [3] for preliminaries and basic results related to monoids and strongly flat acts. First we present some results that we need in the sequel. Proposition Theorem ([2]) Any strongly flat right S -act which has a projective cover is projective. ([2]) Let S be a monoid. The following conditions are equivalent: (i) all right S -acts have strongly flat covers; (ii) S satisfies condition (A), and every cyclic right S -act has a strongly flat cover. For a right congruence σ on a monoid S we define a relation σu by s(σu )t if and only if (us)σ(ut). It is easily checked that σu is also a right congruence on S . 2000 AMS .

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