Related papers: A Note on Noncommutative Even Square Rings
We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that…
We propose the notion of partial resolution of a ring, which is by definition the endomorphism ring of a certain generator of the given ring. We prove that the singularity category of the partial resolution is a quotient of the singularity…
A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this…
In line with known results on groups, we show the existence of the nilpotent radical in Lie rings with minimal condition on centralizers. We also prove a form of Engel's theorem if the characteristic is zero.
In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly oredered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are…
The aim of this article is to introduce the concept of centrally-extended Jordan endomorphisms and proving that if $R$ is a non-commutative prime ring of characteristic not two, and $G$ is a CE- Jordan epimorphism such that $[G(x), x] \in…
In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings…
We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of…
In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also…
The average order of a finite group G is denoted by o(G). In this note, we classify groups whose average orders are less than o(S4), where S4 is the symmetric group on four elements. Moreover, we prove that G \cong S4 if and only if o(G) =…
The target of the present work is to give a new insight in the theory of {\it strongly weakly nil-clean} rings, recently defined by Kosan and Zhou in the Front. Math. China (2016) and further explored in detail by Chen-Sheibani in the J.…
Let $R$ be a ring and let $J(R)$, $C(R)$ be its Jacobson radical and center, correspondingly. If $R$ is a centrally essential ring and the factor ring $R/J(R)$ is commutative, then any minimal right ideal is contained in the center $C(R)$.…
In this paper we discuss several constructions that lead to new examples of nil-clean, clean, and exchange rings. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's…
Set coherence is a basis-independent relational form of quantum coherence: a finite family of quantum states is set incoherent exactly when all its members are diagonal in one common basis. We determine how much low-order Bargmann data are…
In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a…
For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order}…
A ring is said to satisfy the $2$-nil-sum property if every non central-unit is the sum of two nilpotents. We prove that a ring satisfies the $2$-nil-sum property iff it is either a simple ring with the $2$-nil-sum property or a commutative…
In [1], finite associative rings wih identity and such that the set of all zero-divisors form and ideal M, called the Jacobson Radical, of cube zero and square non-zero, were constructed for all the characteristics. These rings are…
A square tiling of the unit square is said to have the minimal tile property if the smallest tile can tile all the other tiles. We show that in such a tiling, the smallest tile cannot be too small.
Let R be an associative ring.In the paper we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n > 2, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is…