On S-torsion exact sequences and Si-projective modules (i = 1,2)

Let [Formula: see text] be a commutative ring and [Formula: see text] a given multiplicative closed subset of [Formula: see text]. In this paper, we introduce the new concept of [Formula: see text]-torsion exact sequences (respectively, [Formula: see text]-torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules.

A bicategorical approach to actions of monoidal categories

We characterize in bicategorical terms actions of monoidal categories on the categories of representations of algebras and of relative Hopf modules. For this purpose we introduce 2-cocycles in any 2-category [Formula: see text]. We observe that under certain conditions the structures of pseudofunctors between bicategories are in one-to-one correspondence with (twisted) 2-cocycles in the image bicategory. In particular, for certain pseudofunctors to Cat, the 2-category of categories, one gets 2-cocycles in the free completion 2-category under Eilenberg–Moore objects, constructed by Lack and Street. We introduce (co)quasi-bimonads in [Formula: see text] and a suitable bicategory of Tambara (co)modules over (co)quasi-bimonads in [Formula: see text] fitting the setting of the latter pseudofuntors. We describe explicitly the involved 2-cocycles in this context and show how they are related to Sweedler’s and Hausser–Nill 2-cocycles in [Formula: see text], which we define. This allows us to recover some results of Schauenburg, Balan, Hausser and Nill for modules over commutative rings. We fit a version of the 2-category of bimonads in [Formula: see text], which we introduced in a previous paper, in a similar setting as above and recover a result of Laugwitz. We observe that pseudofunctors to Cat in general determine what we call pseudo-actions of hom-categories, which correspond to the whole range of a 2-cocycle, so that the described actions of categories appear as restrictions of these 2-cocycles to endo-hom categories.

Factorization invariants of the additive structure of exponential Puiseux semirings

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

Finite generation of the cohomology rings for the extension algebras of monomial algebras

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring [Formula: see text] is finitely generated [Formula: see text] is [Formula: see text] is, where [Formula: see text] is the dual extension of a monomial algebra [Formula: see text] and [Formula: see text] is the opposite algebra of [Formula: see text].

Finite 𝒟C-groups

Let [Formula: see text] be a group and [Formula: see text]. [Formula: see text] is said to be a [Formula: see text]-group if [Formula: see text] is a chain under set inclusion. In this paper, we prove that a finite [Formula: see text]-group is a semidirect product of a Sylow [Formula: see text]-subgroup and an abelian [Formula: see text]-subgroup. For the case of [Formula: see text] being a finite [Formula: see text]-group, we obtain an optimal upper bound of [Formula: see text] for a [Formula: see text] [Formula: see text]-group [Formula: see text]. We also prove that a [Formula: see text] [Formula: see text]-group is metabelian when [Formula: see text] and provide an example showing that a non-abelian [Formula: see text] [Formula: see text]-group is not necessarily metabelian when [Formula: see text]. In particular, [Formula: see text] [Formula: see text]-groups are characterized.

Holomorphic integer graded vertex superalgebras

In this paper, we study holomorphic [Formula: see text]-graded vertex superalgebras. We prove that all such vertex superalgebras of central charge [Formula: see text] and [Formula: see text] are purely even. For the case of central charge [Formula: see text] we prove that the weight-one Lie superalgebra is either zero, of superdimension [Formula: see text], or else is one of an explicit list of 1332 semisimple Lie superalgebras.

On the notion of supplement in acts over monoids

The object of this paper is to generalize the notion of supplement in modules to monoid acts. In contrast to the case of modules that supplements of submodules do not generally exist, here we uniquely characterize the supplement of a proper subact of an act. Supplemented acts are defined as acts whose proper subacts all have proper supplements. We discuss how the property of being supplemented relates to certain other properties of acts. In particular, we prove that being supplemented and being completely reducible coincide.

Rings with S-acc on d-annihilators

A commutative ring [Formula: see text] is said to satisfy acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with [Formula: see text]-acc on d-annihilators, where [Formula: see text] is a multiplicative set. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] satisfies [Formula: see text]-acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is [Formula: see text]-stationary, that is, there exist a positive integer [Formula: see text] and an [Formula: see text] such that for each [Formula: see text] [Formula: see text] We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy [Formula: see text]-acc on d-annihilators. We also study serval properties of rings satisfying [Formula: see text]-acc on d-annihilators. The concept of the amalgamated duplication of [Formula: see text] along an ideal [Formula: see text] [Formula: see text] is studied.

Invertible matrices over a class of semirings

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are “almost idempotent” to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question.

On the regularity of product of irreducible monomial ideals

Let [Formula: see text] be a polynomial ring in [Formula: see text] variables over a field [Formula: see text]. When [Formula: see text], [Formula: see text] and [Formula: see text] are monomial ideals of [Formula: see text] generated by powers of the variables [Formula: see text], it is proved that [Formula: see text]. If [Formula: see text], the same result for the product of a finite number of ideals as above is proved.