Unital Ring Research Articles

Characterizations of clean elements by means of outer inverses in rings and applications

We characterize clean elements in unital and general rings by means of outer inverses. Some special cases, such as both clean and unit-regular elements, or strongly clean elements, are discussed. As an application, we also derive new characterizations of strongly regular elements.

Bass’ Nilpotent Unitary K1-Group of Unitary Ring

Bass’ nilpotent unitary K1-group of the unitary ring is introduced and studied. A set of unitary representatives of this group is found and a complete description of its unipotent representatives is given.

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.

Characterization of Lie multiplicative derivation on alternative rings

In this paper, we generalize the result valid for associative rings Bresar and Martindale III to alternative rings. Let $\mathfrak{R} $ be a unital alternative ring, and $\mathfrak{D} \colon \mathfrak{R} \rightarrow \mathfrak{R} $ is a Lie multiplicative derivation. Then, $\mathfrak{D} $ is the form $\delta + \tau $, where $\delta $ is an additive derivation of $\mathfrak{R} $ and $\tau $ is a map from $\mathfrak{R} $ into its center $\mathcal {\mathfrak{R} }$, which maps commutators into the zero.

Leavitt path algebras over arbitrary unital rings and algebras

We expand the work of Tomforde by further extending the construction of Leavitt path algebras (LPAs) over arbitrary associative, unital rings. We show that many of the results over a commutative ring hold in the more general setting, provide some useful generalizations of prior results, and give a definition for an iterated Leavitt path extension in our context.

The core inverses of differences and products of projections in rings with involution

Let R be a unital ring with involution. The core inverses of differences and products of projections in R are investigated. Let $$p, q\in R$$ be two projections, we obtained equivalent conditions such that $$p-q$$, $$(p-q)^2$$, $$1-qp$$, $$pq+qp$$ and $$(pq-qp)^2$$ are core invertible, respectively. Moreover, expressions for their core inverses are given.

Characterizing finite fields via minimal ring extensions

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that for each ring B such that is an inert (minimal ring) extension. It follows that A is a (finite) field the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes there exists a minimal ring extension such that B is a flat A-module there exists a minimal ring extension such that B is a local ring and is a Galois ring extension.

A “quite superfluous” characterization of the Jacobson radical of an associative ring

It is well-known that the Jacobson radical of a unital ring R is the largest superfluous right ideal of R. While a simple example shows this to be false for non-unital rings, it is shown that the result remains valid in the absence of a unit provided the notion of “superfluous” is taken relative to all regular right ideals. The purpose of this note is to provide a proof for this little-known fact.

Lie centrally metabelian group rings over noncommutative rings

Let R be a unital ring and let G be an arbitrary nonabelian group. Lie centrally metabelian group algebras have been studied and described by various authors. In this article, we have classified Lie centrally metabelian group rings over noncommutative rings. We also give characterizations of group rings RG satisfying the Lie identity over both commutative and noncommutative rings.

Simplicity of Ore monoid rings

Given a non-associative unital ring R, a monoid G and a set π of additive maps R→R, we introduce the Ore monoid ring R[π;G], and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids.

Isomorphism invariants for linear quasigroups

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.

Simple flat Leavitt path algebras are von Neumann regular

For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40 years ago who called such rings SF-rings (i.e. simple modules are flat). In this note we show that an SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids, this implies that the graphs are acyclic. Combining with the Abrams–Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi’s question in positive for the class of Leavitt path algebras.

Some products of commutators in an associative ring

Let [Formula: see text] be a unital associative ring and let [Formula: see text] be the two-sided ideal of [Formula: see text] generated by all commutators [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] [Formula: see text]. It has been known that, if either [Formula: see text] or [Formula: see text] is odd, then [Formula: see text] for all [Formula: see text]. This was proved by Sharma and Srivastava in 1990 and independently rediscovered later (with different proofs) by various authors. The aim of our paper is to give a simple proof of the following result: if at least one of the integers [Formula: see text] is odd, then for all [Formula: see text], [Formula: see text] Since it has been known that, in general, [Formula: see text] our result cannot be improved further for all [Formula: see text] such that at least one of them is odd.

Using the Steinberg algebra model to determine the center of any Leavitt path algebra

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subset is compact if and only if its associated hereditary and saturated set of vertices satisfies Condition (F). We also give a basis of the center. Its cardinality depends on the number of minimal compact open invariant subsets of the unit space.

Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in \(\mathrm {SL}_n(R)\) is a product of elementary matrices with entries in R. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where R is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in \(\mathrm {GL}_n(R)\) as a product of exponentials.

A new class of nil-clean elements which are exchange

An element a in a ring R is called left medium nil-clean if $$ a=e+t $$ with idempotent e and nilpotent t, such that et and t commute. This proper class of nil-clean elements properly includes the strongly nil-clean elements. We show that in any unital ring, left medium nil-clean elements are exchange. Over projective-free domains we show that $$2\times 2$$ and $$3\times 3$$ left (or right) medium nil-clean matrices are strongly nil-clean.

FINITE UNITARY RINGS IN WHICH ALL SYLOW SUBGROUPS OF THE GROUP OF UNITS ARE CYCLIC

We characterise finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^{\ast }$ are cyclic. To be precise, we show that, up to isomorphism, $R$ is one of the three types of rings in $\{O,E,O\oplus E\}$ , where $O\in \{GF(q),\mathbb{Z}_{p^{\unicode[STIX]{x1D6FC}}}\}$ is a ring of odd cardinality and $E$ is a ring of cardinality $2^{n}$ which is one of seven explicitly described types.

Sofic mean length

Given a length function L on the R-modules of a unital ring R, for each sofic group Γ we define a mean length for every locally L-finite RΓ-module relative to a bigger RΓ-module. We establish an addition formula for the mean length.We give two applications. The first one shows that for any unital left Noetherian ring R, RΓ is stably direct finite. The second one shows that for any ZΓ-module M, the mean topological dimension of the induced Γ-action on the Pontryagin dual of M coincides with the von Neumann–Lück rank of M.

Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings

Let C be a chain complex of finitely generated free modules over a commutative Laurent polynomial ring Ls in s indeterminates. Given a group homomorphism p:Zs➝Zt we let p!(C)=C⊗LsLt denote the resulting induced complex over the Laurent polynomial ring Lt in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that bk(p!(C))>bk(C), have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings.

Mobi algebra as an abstraction to the unit interval and its comparison to rings

We introduce a new algebraic structure, called mobi algebra, consisting of three constants and one ternary operation. The canonical example of a mobi algebra is the unit interval with the three constants 0, 1, and and the ternary operation given by the formula . We study some of its properties and prove that every unitary ring with one half uniquely determines and is uniquely determined by a mobi algebra with one double. Another algebraic structure, called involutive medial monoid (IMM), is considered to establish the passage between rings and mobi algebras.