Search

Mock‐Lie algebras, a unique class of commutative algebras that satisfy the Jacobi identity, are gaining attention for their potential applications in various mathematical contexts. This study presents a matrix representation of the centroid for Mock‐Lie algebras in dimensions up to four, offering insights into their structure and interactions. By applying the centroid concept and utilizing matrix operations, we explore the relationships between matrix elements, leading to a better understanding of these algebras. Our findings aim to contribute to the growing interest in Mock‐Lie algebras and their role within the broader algebraic landscape. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unit (respectively, unitary Cayley) graph of R is a simple graph G(R) (respectively, Γ(R)) with vertex set R, where two distinct vertices x and y are adjacent if and only if x + y (respectively, x − y) is a unit of R. In this paper, we explore when G(R) and Γ(R) are isomorphic for a finite ring R. Among other results, we prove that G(R) is isomorphic to Γ(R) for a finite ring R if and only if char(R/J(R))=2 or R/J(R)≅ℤ2 × S, where J(R) is the Jacobson radical of R and S is a finite ring. [ABSTRACT FROM AUTHOR]

An element a of a ring R is called a quasi-idempotent if a 2 = k a for some central unit k of R , or equivalently, a = k e , where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to ℤ 2 ; For an indecomposable commutative semilocal ring R with at least two maximal ideals, n (R) (n ≥ 2) is strongly quasi-clean if and only if n (R) is quasi-clean if and only if min { | R / m | , m is a maximal ideal of R } > n + 1. For a prime p and a positive integer n ≥ 2 , n (ℤ (p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed. [ABSTRACT FROM AUTHOR]

Let ℚ be the rational filed. For a square-free integer d with d ≠ 0 , 1 , we denote by K = ℚ (d) the quadratic field. Let K be the ring of algebraic integers of K. In this paper, we completely determine the unit group of the quotient ring K / 〈 ξ n 〉 of K for an arbitrary prime ξ in ℚ (d) , where ℚ (d) has the unique factorization property, and n > 1 is a rational integer. [ABSTRACT FROM AUTHOR]

AbstractAn element a in a ring R is called a quasi-idempotent if a2=ka for some central unit k of R. A ring R is called quasi-exchange if for any element x of R, there exists a∈R with a2=ka∈Rx and k−a∈R(k−x) , where k is a central unit of R. Quasi-exchange rings are a generalization of exchange rings and quasi-clean rings. This paper studies the properties of quasi-exchange rings and the relationship among quasi-exchange rings, exchange rings and quasi-clean rings. An equation xa−fx=1 is used to characterize the quasi-exchange ring and we prove that quasi-exchange rings are left-right symmetric. [ABSTRACT FROM AUTHOR]

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let R be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph Γ (R) , the compressed zero-divisor graph Γ E (R) and the unit graph G (R). We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of Γ (R) and Γ (R [ x ; α , δ ]) , as well as the relations between the domination number of G (R) and G (R [ [ x ; α ] ]) , are studied. [ABSTRACT FROM AUTHOR]

Motivated by Hirano-Tominaga's work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices. [ABSTRACT FROM AUTHOR]

Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with (G(R)) = 3 or 4. [ABSTRACT FROM AUTHOR]