Unital Ring Research Articles

On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.

On the Generalized Strongly Nil-Clean Property of Matrix Rings

Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].

Isomorphisms and derivations of partial flag incidence algebras

Let [Formula: see text] and [Formula: see text] be finite posets and [Formula: see text] a commutative unital ring. In the case where [Formula: see text] is indecomposable, we prove that the [Formula: see text]-linear isomorphisms between partial flag incidence algebras [Formula: see text] and [Formula: see text] are exactly those induced by poset isomorphisms between [Formula: see text] and [Formula: see text]. We also show that the [Formula: see text]-linear derivations of [Formula: see text] are trivial.

Generalised reduced modules

Let $R$ be a commutative unital ring, $a\in R$ and $t$ a positive integer. $a^{t}$-reduced $R$-modules and universally $a^{t}$-reduced $R$-modules are defined and their properties given. Known (resp. new) results about reduced $R$-modules are retrieved (resp. obtained) by taking $t=1$ and results about reduced rings are deduced.

The Zariski covering number for vector spaces and modules

Given a module M over a commutative unital ring R, let denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers M; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare–Tikaradze [Comm. Algebra, in press] showed in several cases that where SM is the set of maximal ideals with Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose ‘some’ residue fields are infinite, which we call the Zariski covering number To do so, we first define the “induced Zariski topology” τ on M, and now define to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result: for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) whenever is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities again become equalities when the image of M under the continuous map (with appropriate Zariski-type topologies) is a κM -Baire subspace of the product space.

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

Abstract Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

Parabolic induction via the parabolic pro-𝑝 Iwahori–Hecke algebra

Let G \mathbf {G} be a connected reductive group defined over a locally compact non-archimedean field F F , let P \mathbf {P} be a parabolic subgroup with Levi M \mathbf {M} and compatible with a pro- p p Iwahori subgroup of G ≔ G ( F ) G ≔\mathbf {G}(F) . Let R R be a commutative unital ring. We introduce the parabolic pro- p p Iwahori–Hecke R R -algebra H R ( P ) \mathcal {H}_R(P) of P ≔ P ( F ) P ≔\mathbf {P}(F) and construct two R R -algebra morphisms Θ M P : H R ( P ) → H R ( M ) \Theta ^P_M\colon \mathcal {H}_R(P)\to \mathcal {H}_R(M) and Ξ G P : H R ( P ) → H R ( G ) \Xi ^P_G\colon \mathcal {H}_R(P) \to \mathcal {H}_R(G) into the pro- p p Iwahori–Hecke R R -algebra of M ≔ M ( F ) M ≔\mathbf {M}(F) and G G , respectively. We prove that the resulting functor Mod ⁡ - H R ( M ) → Mod ⁡ - H R ( G ) \operatorname {Mod}\text {-}\mathcal {H}_R(M) \to \operatorname {Mod}\text {-}\mathcal {H}_R(G) from the category of right H R ( M ) \mathcal {H}_R(M) -modules to the category of right H R ( G ) \mathcal {H}_R(G) -modules (obtained by pulling back via Θ M P \Theta ^P_M and extension of scalars along Ξ G P \Xi ^P_G ) coincides with the parabolic induction due to Ollivier–Vignéras. The maps Θ M P \Theta ^P_M and Ξ G P \Xi ^P_G factor through a common subalgebra H R ( M , G ) \mathcal {H}_R(M,G) of H R ( G ) \mathcal {H}_R(G) which is very similar to H R ( M ) \mathcal {H}_R(M) . Studying these algebras H R ( M , G ) \mathcal {H}_R(M,G) for varying ( M , G ) (M,G) we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction.

On the coefficient-choosing game

Nora and Wanda are two players who choose coefficients of a degree $d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington and Zbarsky to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to $3$ or greater than $8$) with no roots in the maximal abelian extension of $\mathbb{Q}$.

Weak core inverses and pseudo core inverses in a ring with involution

In 2020, Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the m-weak group inverse. Furthermore, we prove that an element is pseudo core invertible if this element is both {1, 3}-invertible and weak group invertible. Then, we show that each nilpotent element is Moore–Penrose invertible if and only if each pseudo core invertible element is weak core invertible. Finally, we consider the conditions under which the weak core inverse of an element coincides with another generalized inverse.

Real Spectrum Versus ℓ-Spectrum via Brumfiel Spectrum

It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We complete the existing list of containments and non-containments between the associated spectral spaces and their spectral subspaces, by proving the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ℓ-spectrum can be embedded, as a spectral subspace, into a real spectrum. The commutative unital rings and Abelian lattice-ordered groups constructed in Eqs. 2, 3, and 4 all have cardinality ℵ1. Moreover, Eq. 3 solves a problem stated in 2012 by Mellor and Tressl.

A result related to derivations on unital semiprime rings

The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.

Gabriel localization in functor categories

Gabriel showed that for a unital ring R, there exists a bijective correspondence between the set of Gabriel filters of R and the set of Giraud subcategories of In this paper we prove a result analogous to the one given by Gabriel: for a small preadditive category there exists a bijective correspondence between the left Gabriel filters of and Giraud subcategories of

Simplicity of iterated Ore extensions

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

Nonadditive commuting maps of unital rings with idempotents

Let 𝒜 be any ring. A nonadditive map φ:𝒜→𝒜 is said to be commuting if [φ(a),b]=[a,φ(b)] holds for all a,b∈𝒜, and is of a standard form if φ(a)=za+f(a) for all a∈𝒜, where z is in the center of 𝒜 and f is a map from 𝒜 into its center. In this paper, we give a complete characterization of φ on unital rings 𝒜 with a nontrivial idempotent. It is also shown that, under some suitable assumptions on 𝒜, φ has a standard form. As applications, the structures of such maps on some important operator algebras are obtained.

Jacobson’s Lemma in the ring of quaternionic linear operators

Abstract In the present paper, we study the Jacobson’s Lemma in the unital ring of all bounded right linear operators ℬ R (X) acting on a two-sided quaternionic Banach space X. In particular, let A, B ∈ ℬ R (X) and let q ∈ ℍ \ {0}, we prove that w(AB) \ {0} = w(BA) \ {0} where w belongs to the spherical spectrum, the spherical approximate point spectrum, the right spherical spectrum, the left spherical spectrum, the spherical point spectrum, the spherical residual spectrum and the spherical continuous spectrum. We also prove that the range of (AB)2 − 2Re(q)AB + |q|2 I is closed if and only if (BA)2 − 2Re(q)BA + |q|2 I has closed range. Finally, we show that (AB)2 − 2Re(q)AB + |q|2 I is Drazin invertible if and only if (BA)2 − 2Re(q)BA + |q|2 I is.

Isonoetherian power series rings II

We study the isonoetherian and “ACC d on ideals” properties in power series rings, polynomial rings, rings of the form (where I is an ideal of a commutative unitary ring R), (where D is an integral domain with quotient field K), Nagata’s idealization ring and almost pseudo valuation domain. We point out three mistakes in two works. Let R be a quasi-local proper subring of an integral domain S such that R is integrally closed in S. We prove that is u-isonoetherian if and only if R satisfies ACC d on ideals, S is Noetherian and there exists a maximal ideal M of S that is a divided prime ideal of R and R/M is a rank one discrete valuation domain with quotient field S/M. We prove that the ring is isonoetherian if and only if R is a rank one discrete valuation domain with quotient field S. We deduce an example of a u-isonoetherian ring extension such that the ring is not isonoetherian.

On Unit Fine Rings

A nonzero element a in a unital ring R is called unit fine if there is a unit u in R such that ua is fine (i.e. a sum of a unit and a nilpotent). A ring all whose elements are unit fine is called accordingly. This turns out to be a new class of simple rings, including the class of fine rings (and so the class of simple Artinian rings). The paper studies unit fine elements and rings. For rings such that 1 is a sum of two units, it is proved that matrix rings over unit fine rings, are unit fine.

Strongly graded Leavitt path algebras

Let [Formula: see text] be a unital ring, let [Formula: see text] be a directed graph and recall that the Leavitt path algebra [Formula: see text] carries a natural [Formula: see text]-gradation. We show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.

Kitaev’s Stabilizer Code and Chain Complex Theory of Bicommutative Hopf Algebras

In this paper, we give a generalization of Kitaev’s stabilizer code based on chain complex theory of bicommutative Hopf algebras. Due to the bicommutativity, the Kitaev’s stabilizer code extends to a broader class of spaces, e.g. finite CW-complexes ; more generally short abstract complex over a commutative unital ring R which is introduced in this paper. Given a finite-dimensional bisemisimple bicommutative Hopf algebra with an R-action, we introduce some analogues of $$\mathbb {A}$$ -stabilizers, $$\mathbb {B}$$ -stabilizers and the local Hamiltonian, which we call by the $$(+)$$ -stabilizers, the $$(-)$$ -stabilizers and the elementary operator respectively. We prove that the eigenspaces of the elementary operator give an orthogonal decomposition and the ground-state space is isomorphic to the homology Hopf algebra. In application to topology, we propose a formulation of topological local stabilizer models in a functorial way. It is known that the ground-state spaces of Kitaev’s stabilizer code extends to Turaev-Viro TQFT. We prove that the 0-eigenspaces of a topological local stabilizer model extends to a projective TQFT which is improved to a TQFT in typical examples. Furthermore, we give a generalization of the duality in the literature based on the Poincare-Lefschetz duality of R-oriented manifolds.

Structure of centralizer matrix algebras

Given an n×n matrix c over a unitary ring R, the centralizer of c in the full n×n matrix ring Mn(R) is called a principal centralizer matrix ring, denoted by Sn(c,R). We investigate its structure and prove: (1) If c is an invertible matrix with a c-free point, or if R has no zero-divisors and c is a Jordan-similar matrix with all eigenvalues in the center of R, then Mn(R) is a separable Frobenius extension of Sn(c,R) in the sense of Kasch. (2) If R is an integral domain and c is a Jordan-similar matrix, then Sn(c,R) is a cellular R-algebra in the sense of Graham and Lehrer. In particular, if R is an algebraically closed field and c is an arbitrary matrix in Mn(R), then Sn(c,R) is always a cellular algebra, and the extension Sn(c,R)⊆Mn(R) is always a separable Frobenius extension.