Unital Ring Research Articles

A characterization of ramification groups via jet algebras

AbstractWe present a functorial method to define ramification groups, identifying them as inertia groups of an induced action on composite jet algebras. This framework lays the foundation for defining higher ramification groups for actions involving group schemes. To achieve this, we introduce Taylor maps within the category of commutative unitary rings at prime ideals of an $$R$$ R -algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.

Rings with un − 1 nilpotent for each unit u

We continue the study in-depth of the so-called [Formula: see text]-UU rings for any [Formula: see text], that were defined by the first-named author in [Danchev, On exchange P-UU unital rings, Toyama Math. J. 39(1) (2017) 1–7] as those rings [Formula: see text] for which [Formula: see text] is always a nilpotent for every unit [Formula: see text]. Specifically, for any [Formula: see text], we prove that a ring is strongly [Formula: see text]-nil-clean if, and only if, it is simultaneously strongly [Formula: see text]-regular and an [Formula: see text]-UU ring. This somewhat extends results due to [Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211; Abyzov, Strongly q-nil-clean rings, Sib. Math. J. 60(2) (2019) 197–208; Cui and Danchev, Some new characterizations of periodic rings, J. Algebra Appl. 19(12) (2020)]. Moreover, our results somewhat improves the ones obtained by Koşan et al. [Rings such that, for each unit [Formula: see text], [Formula: see text] belongs to the Jacobson radical, Hacettepe J. Math. Stat. 49(4) (2020) 1397–1404].

The lattice of ideals of certain rings

Let A be a unitary ring and let (I(A),⊆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathbf {I(A)},\\subseteq )$$\\end{document} be the lattice of ideals of the ring A. In this article, we will study the property of the lattice (I(A),⊆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathbf {I(A)},\\subseteq )$$\\end{document} to be Noetherian or not, for various types of rings A. In the last section of the article, we study certain rings that are not Boolean rings, not fields, but all their ideals are idempotent.

On the equation $a^{2}x=a$ in unital rings

In an arbitrary ring, the equation in the title defines the left strongly regular elements. Elements that are both left and right strongly regular are simply called strongly regular. If all elements of a ring are left strongly regular, then they are, in fact, strongly regular and this is the definition of strongly regular rings. We provide a characterization of when a left strongly regular element is indeed strongly regular, based on an intrinsic condition. While we show that it is not possible to give $2\times 2$ non-examples over $\mathbb{Z}$ or in certain matrix rings over $\mathbb{Z}_{n}$ for $n\in \{8,9,16\}$, we present two examples by George Bergman: a left strongly regular element that is not regular and a regular, left strongly regular element that is not strongly regular. Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take N≤2 for matrix rings, and that one may choose N≤3 for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that N≤6 can be arranged.For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

Groups with 𝖠_{ℓ}-commutator relations

If A A is a unital associative ring and ℓ ≥ 2 \ell \geq 2 , then the general linear group GL ⁡ ( ℓ , A ) \operatorname {GL}(\ell , A) has root subgroups U α U_\alpha and Weyl elements n α n_\alpha for α \alpha from the root system of type A ℓ − 1 \mathsf A_{\ell - 1} . Conversely, if an arbitrary group has such root subgroups and Weyl elements for ℓ ≥ 4 \ell \geq 4 satisfying natural conditions, then there is a way to recover the ring A A . A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring M ⁡ ( ℓ , A ) , \operatorname {M}(\ell , A), a nonunital associative ring with a well-behaved Peirce decomposition is provided.

On center graphs of finite associative rings

UDC 512.5 We consider a finite associative ring R , which may have or may not have a unit element. We also examine its center denoted by Z ( R ) . Our main focus is on the introduction of two distinct graphs associated with R , namely, the center graph denoted by G C ( R ) and the strict center graph denoted by G C ( R ) ¯ . We present the properties of G C ( R ) and explore its implications on the nature of Z ( R ) . Specifically, we demonstrate that if G C ( R ) is complete, then Z ( R ) is an ideal in R . Furthermore, in the case where R is a unital ring, the completeness of G C ( R ) leads to the conclusion that R is a commutative ring. As a specific application of our results, we provide an explicit construction of the graph G C ¯ ( T 2 ( p ) ) , where T 2 ( p ) represents the ring of upper-triangular matrices with entries in the ring ℤ / p ℤ and p is a prime integer. In our investigations of the center graph and strict center graph, we aim to shed light on the properties of finite associative rings and their centers, thus providing valuable insights and applications in the ring theory.

Computing the closure of a support

When E is an R-module over a commutative unital ring R, the Zariski closure of its support is of the form V ( O ( E ) ) where O ( E ) is a unique radical ideal. We give an explicit form of O ( E ) and study its behavior under various operations of algebra. Applications are given, in particular for ring extensions of commutative unital rings whose supports are closed. We provide some applications to crucial and critical ideals of ring extensions.

A Note on Abelian Groups Supporting Unital Rings

A Note on Abelian Groups Supporting Unital Rings

Very good gradings on matrix rings are epsilon-strong

We investigate properties of group gradings on matrix rings M n ( R ) , where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on M n ( R ) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that M n ( R ) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when M n ( R ) is an epsilon-crossed product. Our results are illustrated by several examples.

On left annihilating content polynomials and power series

Let R be an associative unital ring, and let f ∈ R [ x ] . We say that f is a left annihilating content (AC) polynomial if f = af 1 for some a ∈ R and f 1 ∈ R [ x ] with l R [ x ] ( f 1 ) = 0 . The ring R is called a left EM-ring if each f ∈ R [ x ] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a ∈ R and a finitely generated right ideal J of R with l R ( J ) = 0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R [ G ] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R [ x ] , R [ x , x − 1 ] , and R [ [ x ] ] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings).

Rings of finite Morley rank without the canonical base property

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.

The Build-Up Construction for Codes over a Commutative Non-Unitary Ring of Order 9

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes completely in length at most 3, and partially in lengths 4 and 5, up to monomial equivalence.

GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS

Abstract If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$ . Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property.

On the Construction of a Groupoid from an Ample Hausdorff Groupoid with Twisted Steinberg Algebra not Isomorphic to its Non-twisted Steinberg Algebra

This study introduces an ample Hausdorff groupoid $\hat{A} \rtimes \mathcal{R}$ extracted from an ample Hausdorff groupoid $\mathcal{G}$ and a unital commutative ring $R$; a Hausdorff groupoid $D$ which is the discrete twist over $\hat{A} \rtimes \mathcal{R}$. In the groupoid C*-algebra perspective, when $R = \mathbb{C}$ there is an isomorphism between the non-twisted groupoid C*-algebra $(C^*(\mathcal{G}))$ and the twisted groupoid C*-algebra $(C^*(\hat{A} \rtimes \mathcal{R};D))$. However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra $(A_R(\mathcal{G}))$ and the twisted Steinberg algebra $(A_R(D; \hat{A} \rtimes \mathcal{R}))$ are non-isomorphic.

Applications of reduced and coreduced modules I

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.

Characterizing Rickart and Baer ultragraph Leavitt path algebras

We characterize ultragraph Leavitt path algebras that are Rickart, locally Rickart, graded Rickart, and graded Rickart *-rings. We also characterize ultragraph Leavitt path algebras that are Baer, locally Baer, graded Baer, Baer *-rings, and combinations of these. These characterizations build on and generalize the work of Hazrat and Vaš on Leavitt path algebras over fields to ultragraph Leavitt path algebras over semi-simple commutative unital rings.

The build up construction for codes over a non-commutative non-unitary ring of order $ 9 $

<abstract><p>The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order four to generate quasi self-dual codes. In the present paper we introduce three such propagation rules to generate self-orthogonal, one-sided self-dual, and self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes in lengths at most $ 7, $ up to monomial equivalence. Mass formulas for the three classes of codes considered ensure that the classification is complete.</p></abstract>

Lie (Jordan) $ \sigma- $centralizer at the zero products on generalized matrix algebra

<p>Given a unital commutative ring $ \mathscr{R} $, $ (\mathscr{A}, \mathscr{B}) $ and $ (\mathscr{B}, \mathscr{A}) $ are bimodules of $ \mathscr{M} $ and $ \mathscr{N} $, respectively, where $ \mathscr{A}, \mathscr{B} $ are unitals $ \mathscr{R}- $algebras. The $ \mathscr{R}- $algebra $ \mathscr{G} = $ $ \mathscr{G}(\mathscr{A}, \mathscr{M}, \mathscr{N}, \mathscr{B}) $ is a generalized matrix algebra described by the Morita context $ (\mathscr{A}, \mathscr{B}, \mathscr{M}, \mathscr{N}, \zeta_{\mathscr{M}\mathscr{N}}, \chi_{\mathscr{N}\mathscr{M}}) $. The present study investigated the structure of Lie (Jordan) $ \sigma- $centralizers at the zero products on order two generalized matrix algebra and established that each Jordan $ \sigma- $centralizer at the zero products is a $ \sigma- $centralizer at the zero product on order two generalized matrix algebra. We also provided sufficient and necessary conditions under which a Lie $ \sigma- $centralizer at the zero product is proper on an order two generalized matrix algebra.</p>

About unital and non-unital duo rings

About unital and non-unital duo rings