Abstract: Let $\mathcal{C}$ be a finite tensor category with simple unit object, let $\mathcal{Z}(\mathcal{C})$ denote its monoidal center, and let $L$ and $R$ be a left adjoint and a right adjoint of the forgetful functor $U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C}$. We show that the following conditions are equivalent: (1) $\mathcal{C}$ is unimodular, (2) $U$ is a Frobenius functor, (3) $L$ preserves the duality, (4) $R$ preserves the duality, (5) $L(1)$ is self-dual, and (6) $R(1)$ is self-dual, where $1 \in \mathcal{C}$ is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.

Abstract: Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in \mathbb N$, let $\mathscr U_k (H)$ denote the set of all $\ell \in \mathbb N$ with the property that there are atoms $u_1, \ldots, u_k, v_1, \ldots, v_{\ell}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$ (thus, $\mathscr U_k (H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\mathscr U_k (H)$ are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

Abstract: Let M be a monoid. A ring R is called M-π-Armendariz if whenever α = a 1 g 1 + a 2 g 2 + · · · + a n g n , β = b 1 h 1 + b 2 h 2 + · · · + b m h m ∈ R[M] satisfy αβ ∈ nil(R[M]), then a i b j ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.

Abstract: Let F be a field of characteristic p ≥ 0 and G any group. The local nilpotency of the group of units of the group algebra FG is investigated. We show that if 𝒰(FG) is locally nilpotent, then the set of p-elements of G form a subgroup P and the torsion elements of G/P form an abelian group. If, in addition, the set of nilpotent elements of FG is finite, every idempotent in F(G/P) is central; a converse version is also indicated. As a result, we show that, if G is torsion, then 𝒰(FG) is locally nilpotent if and only if G is locally nilpotent and G′ is a p-group, if and only if FG is Lie Engel and G is locally finite.

Abstract: We generalise various non-triviality conditions for group actions to Fell bundles over discrete groups and prove several implications between them. We also study sufficient criteria for the reduced section C*-algebra C_r(B) of a Fell bundle (B_g) to be strongly purely infinite. If the unit fibre A=B_e contains an essential ideal that is separable or of Type I, then the Fell bundle is aperiodic if and only if it is topologically free. If, in addition, G=Z or G=Z/p for a square-free number p, then these equivalent conditions are satisfied if and only if A detects ideals in C_r(B), if and only if A^+ \ {0} supports C_r(B) in the Cuntz sense. For G as above and arbitrary A, C_r(B) is simple if and only if the Fell bundle B is minimal and pointwise outer. In general, B is aperiodic if and only if each of its non-trivial fibres has a non-trivial Connes spectrum. If G is finite or if A contains an essential ideal that is of Type I or simple, then aperiodicity is equivalent to pointwise pure outerness.

Abstract: Let $R$ be a commutative noetherian ring. Denote by $D^-(R)$ the derived category of cochain complexes $X$ of finitely generated $R$-modules with $H^i(X)=0$ for $i\gg0$. Then $D^-(R)$ has the structure of a tensor triangulated category with tensor product $-\otimes_R^L-$ and unit object $R$. In this paper, we study thick tensor ideals of $D^-(R)$, i.e., thick subcategories closed under the tensor action by each object in $D^-(R)$, and investigate the Balmer spectrum $Spc\,D^-(R)$ of $D^-(R)$, i.e., the set of prime thick tensor ideals of $D^-(R)$. First, we give a complete classification of the thick tensor ideals of $D^-(R)$ generated by bounded complexes, establishing a generalized version of the Hopkins-Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum $Spc\,D^-(R)$ and the Zariski spectrum $Spec\,R$, and study their topological properties. After that, we compare several classes of thick tensor ideals of $D^-(R)$, relating them to specialization-closed subsets of $Spec\,R$ and Thomason subsets of $Spc\,D^-(R)$, and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of $D^-(R)$ in the case where $R$ is a discrete valuation ring.

Abstract: Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R2 → R satisfying G(u, u)u = uG(u, u), and G(1, r) = G(r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ(xyx−1y−1) = θ(x)θ(y)θ(x)−1θ(y)−1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism.

Abstract: Let be a field of characteristic and any group. In this article, the Engel property of the group of units of the group algebra is investigated. We show that if is locally finite, then is an Engel group if and only if is locally nilpotent and is a -group. Suppose that the set of nilpotent elements of is finite. It is also shown that if is torsion, then is an Engel group if and only if is a finite -group and is Lie Engel, if and only if is locally nilpotent. If is nontorsion but is semiprime, we show that the Engel property of implies that the set of torsion elements of forms an abelian normal subgroup of .

Abstract: We prove that for a finitely generated linear group G over a field of positive characteristic the family of quotients by finite subgroups has finite asymptotic dimension. We use this to show that the K-theoretic assembly map for the family of finite subgroups is split injective for every finitely generated linear group G over a commutative ring with unit under the assumption that G admits a finite-dimensional model for the classifying space for the family of finite subgroups. Furthermore, we prove that this is the case if and only if an upper bound on the rank of the solvable subgroups of G exists.

Abstract: In this paper, we construct a homeomorphism on the unit closed disk to show that an invertible mapping on a compact metric space is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic.

Abstract: This article considers the category of commutative medial magmas with cancellation, a structure that generalizes midpoint algebras and commutative semigroups with cancellation. In this category each object admits at most one internal monoid structure for any given unit. Conditions for the existence of internal monoids and internal groups, as well as conditions under which an internal reflexive relation is a congruence, are studied.

Abstract: According to Jacobson (31), a right ideal is bounded if it con- tains a non-zero ideal, and Faith (15) called a ring strongly right bounded if every non-zero right ideal is bounded. From (30), a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which sat- isfy Property (A) and the conditions asked by Nielsen (42). It is shown that for a u.p.-monoid M and � : M ! End(R) a compatible monoid homomorphism, if R is reversible, then the skew monoid ring RM is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and � : M ! End(R) is a weakly rigid monoid homomorphism, then the skew monoid ring RM has right Property (A).

Abstract: We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$ , with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$ .

Abstract: Let R be a finite commutative ring with nonzero identity. We define Γ(R) to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of Γ(R) are obtained and the vertex connectivity and the edge connectivity of Γ(R) are given. Finally, by a constructive way, we determine when the graph Γ(R) is Hamiltonian. As a consequence, we show that Γ(R) has a perfect matching if and only if |R| is an even number.

Abstract: The unitary Cayley graph of a ring , denoted , is the simple graph defined on all elements of , and where two vertices and are adjacent if and only if is a unit in . The largest distance between all pairs of vertices of a graph is called the diameter of and is denoted by . It is proved that for each integer , there exists a ring such that . We also show that for a ring with self-injective and classify all those rings with , and , respectively.

Abstract: Let $R$ be a commutative local ring. We study the subcategory of the homotopy category of $R$-complexes consisting of the totally acyclic $R$-complexes. In particular, in the context where $Q\to R$ is a surjective local ring homomorphism such that $R$ has finite projective dimension over $Q$, we define an adjoint pair of functors between the homotopy category of totally acyclic $R$-complexes and that of $Q$-complexes, which are analogous to the classical adjoint pair between the module categories of $R$ and $Q$. We give detailed proofs of the adjunction in terms of the unit and counit. As a consequence, one obtains a precise notion of approximations of totally acyclic $R$-complexes by totally acyclic $Q$-complexes.

Abstract: Applying a theorem according to Rhemtulla and Formanek, we partially solve an open problem raised by Hochman with an affirmative answer. Namely, we show that if G is a countable torsion-free locally nilpotent group that acts by homeomorphisms on X, and S ⊂ G is a subsemigroup not containing the unit of G such that f ∈ 〈 1 , s f : s ∈ S 〉 for every f ∈ C ( X ) , then ( X , G ) has zero topological entropy.

Abstract: Let A' and A" be the dual and bidual spaces of a locally convex algebra A with dual and weak* topology, respectivly. In this paper we show that A has a bounded right (left) approximate identity if and only if A" has a right (left) unit with respect to the first (second) Arens product.

Abstract: An absolute valued algebra is a nonzero real algebra that is equipped with a multiplicative norm (‖xy‖ = ‖x‖‖y‖). We classify, by an algebraic method, all four-dimensional absolute valued algebras containing a nonzero central idempotent. Moreover, we construct a new absolute valued algebras with left unit of four dimension.

Abstract: Given a ring R , a strictly ordered monoid and monoid homomorphism . Constructed the set of all function from S to R whose support is artinian and narrow, with pointwise addition and the skew convolution multiplication, it becomes a ring called the skew generalized power series rings (SGPSR) and denoted by . A ring R is called reduced if it contains no nonzero nilpotent elements, reversible if for all , implies . Let be a ring endomorphism, if for , implies , then is called rigid . If for all , if and only if , then is called compatible. In this paper we will discuss about the constructing of SGPSR homomorphism. Beside that, we also discuss about rigid and compatible endomorphism on SGPSR .

Abstract: We remark that the functor \(\varGamma \) maps a non-equational class of groups, the category of abelian \(\ell \)-groups with strong unit, to an equational class , the variety of all MV-algebras.

Abstract: Thank you very much for downloading unit groups of group rings. As you may know, people have search hundreds times for their chosen readings like this unit groups of group rings, but end up in harmful downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they juggled with some malicious virus inside their computer. unit groups of group rings is available in our digital library an online access to it is set as public so you can download it instantly. Our books collection spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the unit groups of group rings is universally compatible with any devices to read.

Abstract: 1Professor of Psychiatry & Head of the Psychodermatology Unit, School of Medical Sciences, National University of Asunción (Paraguay), 2Professor of Dermatology & Head of the Dermatopathology Unit, School of Medical Sciences, National University of Asunción (Paraguay), Corresponding Author: Email: [email protected] This is an open access journal, and articles are distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License, which allows others to remix, tweak, and build upon the work non-commercially, as long as appropriate credit is given and the new creations are licensed under the identical terms.

Abstract: In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

Abstract: A generalized pseudo effect algebra (GPEA) is a partially ordered partial algebraic structure with a smallest element 0, but not necessarily with a unit (i.e, a largest element). If a GPEA admits a so-called unitizing automorphism, then it can be embedded as an order ideal in its so-called unitization, which does have a unit. We study unitizations of GPEAs with respect to a unitizing automorphism, paying special attention to the behavior of congruences, ideals, and the Riesz decomposition property in this setting.

Abstract: Let $R$ be a commutative ring with identity. A unit $u$ of $R$ is called exceptional if $1-u$ is also a unit. When $R$ is a finite commutative ring, we determine the additive and multiplicative structures of its exceptional units; and then as an application we find a necessary and sufficient condition under which $R$ is generated by its exceptional units.