Some homological characterizations of semigroups and semirings

In this paper we characterize semirings all of whose p-injective semimodules are injective. We also classify monoids all of whose p-injective acts are injective.     

Absolutely f-Equationally Compact Monoids

This note describes monoids over which all left acts are f-equationally compact (equivalently, 1-equationally compact). It is also proved that every commutative absolutely 1-pure absolutely 1-equationally compact monoid is injective.     

Projectivity of acts and morita equivalence of monoids

In this paper we shall consider a non-additive category of A-modules, that is, instead of a ring A we take a monoid A which acts on sets from the left. These objects will be called A-acts. We investigate indecomposable A-acts and generators and characterize projectives in this category. For a given monoid A we describe all monoids B such that the category of B-acts is equivalent to the category of A-acts. In particular we find that equivalence of these categories yields an isomorphism between the monoids A and B if A is a group or finite or commutative. This differs from the additive case where the categories of modules over a commutative field and its ring of nxn matrices are equivalent. Finally we give examples of non-isomorphic monoids A and B such that the corresponding categories are equivalent.     

Monoids for Which Condition (P) Acts are Projective

A characterisation of monoids for which all right S-acts satisfying conditions (P) are projective is given. We also give a new characterisation of those monoids for which all cyclic right S-acts satisfying condition (P) are projective, similar in nature to recent work by Kilp [6]. In addition we give a sufficient condition for all right S-acts that satisfy condition (P) to be strongly flat and show that the indecomposable acts that satisfy condition (P) are the locally cyclic acts.     

A note on the structure of injective diagrams

We determine all indecomposable injective diagrams over a poset I with values in some category R-Mod of modules and characterize the case when every injective diagram decomposes into a direct sum of indecomposables. Moreover, we show that injective diagrams have a standard form dual to that of projective diagrams iff I is a noetherian poset.     

Conjugacy of idempotents in jordan pairs

The injectives in the category of associative rings and homomorphisms with accessible images are investigated.It is shown that every such ring has an identity and is a direct sum of finitely many indecomposable injectives.The indecomposable injectives are shown to be simple when they are algebras over fields other than Z_p and in the remaining characteristic p case to have only three ideals.A necessary and sufficient condition is obtained for certain finite direct sums of indecomposable injectives to be injective.     

On strongly (<Emphasis Type="Italic">P</Emphasis>)-cyclic acts

By a regular act we mean an act such that all its cyclic subacts are projective. In this paper we introduce strong (P)-cyclic property of acts over monoids which is an extension of regularity and give a classification of monoids by this property of their right (Rees factor) acts.     

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.     

Axiomatizability of the class of weakly injective <Emphasis Type="Italic">S</Emphasis>-acts

The concepts of weakly injective, fg-weakly injective, and p-weakly injective S-acts generalize that of injective S-act. We study the monoids S over which the classes of weakly injective, fg-weakly injective, and p-weakly injective S-acts are axiomatizable. We prove that the class of p-weakly injective S-acts over a regular monoid is axiomatizable.     

The Monoid of All Injective Order Preserving Partial Transformations on a Finite Chain

In this paper we study several structural properties of the monoids \poi _n of all injective order preserving partial transformations on a chain with n elements. Our main aim is to give a presentation for these monoids. January 27, 1999     

A note on three artinian rings

A module over an artinian ring is uniserial if it has a unique composition series, and an artinian ring is serial if each of its indecomposable projective modules is uniserial. Fuller [4, Theorem 5.4] showed that an artinian ring R is serial if and only if each of left indecomposable projective and injective R-modules is uniserial. The following question was raised in 4, p.134: Is an artinian ring R necessarily serial if each of its indecomposable injective modules is uniserial? Example 1 in this note answers this question in the negative     

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.     

C(P)系对幺半群的刻画

设S是幺半群.本文介绍并研究了正则右系的一个推广.一个右S-系A称为C(P)系,如果A的所有循环子系满足条件(P).本文证明了右C(P)系形成了右S-系的一个新的类,同时,C(P)性质为幺半群同调分类研究提供了新思路.     

A Generalization of Regular Left Acts

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Quasipure injective torsion-free groups with indecomposable pure subgroups

In a quasipure injective torsion-free Abelian group whose pure subgroups are strongly indecomposable, any nonzero endomorphism is shown to be a monomorphism. The results of this paper together with results obtained earlier describe quasipure injective torsion-free groups. As a corollary, it is proved that any quasipure injective torsion-free group is transitive. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 587–592, October, 2000.     

On morphisms of commutative monoids

The aim of this work is to study monoid morphisms between commutative monoids. Algorithms to check if a monoid morphism between two finitely generated monoids is injective and/or surjective are given. The structure of the set of monoid morphisms between a monoid and a cancellative monoid is also studied and an algorithm to obtain a system of generators of this set is provided.     

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.     

Quotients of Bing spaces

A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces.     

The Cartan matrix of a centralizer algebra

The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension.