Related papers: On partial augmentations of elements in integral g…
Based on the notion of a $\Delta$-group(oid), ring-valued invariants of pairs of topological spaces can be defined in intrinsic topological terms.
We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is…
Orthogonal sets of idempotents are used to design sets of unitary matrices, known as constellations, such that the modulus of the determinant of the difference of any two distinct elements is greater than $0$. It is shown that unitary…
In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a…
A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is…
In what follows we generalize the notion of a complemented ring to rings that are not necessarily reduced. We then determine how our concepts fit in with other well-known classes of rings.
A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative…
The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither…
In this paper, we study partial actions of groups on $R$-algebras, where $R$ is a commutative ring. We describe the partial actions of groups on the indecomposable algebras with enveloping actions. Then we work on algebras that can be…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants…
In this paper, we investigate the structural and characterizing properties of the so-called {\it 2-UQ rings}, that are rings such that the square of every unit is the sum of an idempotent and a quasi-nilpotent element that commute with each…
The aim of this article is to draw attention towards various natural but unanswered questions related to the lower central series of the unit group of an integral group ring.
We develop an elementary theory of partially additive rings as a foundation of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of classical algebraic geometry is established very straightforwardly. As applications, (1)…
We give a comprehensive study of the so-called \textit{semi-tripotent rings} obtaining their new and non-trivial characterization as well as a complete description in terms of sums and products of some special elements. Particularly, we…
This paper establishes mixed multiplicity formulas concerning the relationship between mixed multiplicities of modules and mixed multiplicities of rings via rank of modules.
In order to study the properties of SEP elements, we propose the concepts of one sided a_idempotent and one sided a_equivalent. Under the condition that an element in a ring is both group invertible and MP_invertible, some equivalent…
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…
It is well known that there exist non-isomorphic compact groups with isomorphic representation rings (fusion rules). Nevertheless, considerable structural information about the group can be reconstructed from its representation ring. We…
We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.
Some general properties of abstract relations are closely examined. These include generalizations of linearity, and properties based on `pinning' an inequality by a pair of families of endomorphisms.To each property we try to associate a…