Monoids over which all strongly flat left acts are regular

Let Sbe a monoid. It is shown that all strongly flat left S-acts are regular if and only if all left S-acts having the property (E) are regular if and only if Sis a left PP monoid and satisfies (FP₂).This result answers a question in Kilp and Knauer [5].     

On covers of cyclic acts over monoids

In (Bull. Lond. Math. Soc. 33:385–390, 2001) Bican, Bashir and Enochs finally solved a long standing conjecture in module theory that all modules over a unitary ring have a flat cover. The only substantial work on covers of acts over monoids seems to be that of Isbell (Semigroup Forum 2:95–118, 1971), Fountain (Proc. Edinb. Math. Soc. (2) 20:87–93, 1976) and Kilp (Semigroup Forum 53:225–229, 1996) who only consider projective covers. To our knowledge the situation for flat covers of acts has not been addressed and this paper is an attempt to initiate such a study. We consider almost exclusively covers of cyclic acts and restrict our attention to strongly flat and condition (P) covers. We give a necessary and sufficient condition for the existence of such covers and for a monoid to have the property that all its cyclic right acts have a strongly flat cover (resp. (P)-cover). We give numerous classes of monoids that satisfy these conditions and we also show that there are monoids that do not satisfy this condition in the strongly flat case. We give a new necessary and sufficient condition for a cyclic act to have a projective cover and provide a new proof of one of Isbell’s classic results concerning projective covers. We show also that condition (P) covers are not unique, unlike the situation for projective covers.     

On free,projective, and strongly flat acts

Archiv der Mathematik -     

On monoids over which all generators satisfy a flatness property

This paper addresses conditions under which all generators in the category of right S-acts (where S is a monoid) satisfy a flatness property. There are characterizations for monoids over which all generators satisfy a flatness property α where α can stand for freeness, projectivity, strong flatness, Condition (P), principal weak flatness and torsion freeness. To our knowledge, the problem has not been studied for other flatness properties such as weak flatness, Condition (E) and regularity. The present paper addresses this gap.     

Hereditary endomorphism monoids of projective acts

Projective acts whose endomorphism monoids are left or right (semi-) hereditary are characterized. For example, it is shown that for a noncyclic free or projective S-act P, End P is left (semi) hereditary if and only if P ≈ Se₁ Π Se₂ and e₁Se₁, e₂Se₂ are groups.     

Endomorphism monoids of acts over monoids

In this paper we investigate under which conditions a monoid R is defined by the endomorphism monoid of an act over R. More precisely, we ask when an isomorphism between two such endomorphism monoids over monoids R₁ and R₂ is induced by a semilinear isomorphism. The question is considered also for ordered and for topological monoids. On the way we characterize monoids over which all projective acts are free. An abstract of this paper appeared in the Proceedings of the Conference on Semigroups, Szeged 1972.     

On Rees weakly projective right acts

In this article, we present an extension of the concept of weakly projective acts to the socalled (S/I, S/J)-projective acts. Retracts, coproducts, and products of acts as well as Rees factor acts are considered from the point of view of these properties. They are used to describe monoids that are disjoint unions of a group with a left zero semigroup or with a disjoint union of simple right ideals. We suggest concepts of weak QF-monoids. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 85–96, 2004.     

Description of commutative monoids over which all polygons areW-stable

Translated from Algebra i Logika, Vol. 28, No. 4, pp. 371–381, July–August, 1989.     

Covers of acts over monoids II

In 1981 Edgar Enochs conjectured that every module over a unitary ring has a flat cover. He finally proved this conjecture in 2001, in a paper that included an independent proof by Bican and El Bashir. Enochs had in fact considered different types of covers as early as 1963, for example injective and torsion free covers, and since then a great deal of effort has been spent on their study. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the ‘other’ type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective covers and demonstrate that in some cases the results are quite different from the module case.     

Rings over which all modules are semiregular

For a ring A, it is proved that all A-modules are semiregular if and only if A is an Artinian serial ring and J ²(A) = 0. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 185–194, 2007.     

Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).     

On left absolutely flat monoids

Communicated by J. M. Howie