Related papers: Units in Near-rings
We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in…
A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}.…
The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.
We show that there exist zero-symmetric simple nearrings with identity which are not equiprime, solving a longstanding open problem.
We define the concepts of weakly precious and precious rings which generalize the notions of weakly clean and nil-clean rings. We obtain some fundamental properties of these rings. We also obtain certain subclasses of such rings and then…
We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules,…
In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions…
We investigate the notion of \textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean NI ring,…
Let p be an odd prime, F the field of p elements and G a finite abelian p-group with an arbitrary involutory automorphism. Extend this automorphism to the group algebra FG and consider the unitary and the symmetric normalized units of FG.…
A near permutation of a set is a bijection between two cofinite subsets, modulo coincidence on smaller cofinite subsets. Near permutations of a set form its near symmetric group. In this monograph, we define near actions as homomorphisms…
We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings, and construct examples of such rings…
We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.
In this paper, we present the Nullstellensatz in case of the coordinate rings of a nonempty subset of Kn where K is a finite field Fq. Some applications of the Nullstellensatz are also discussed.
We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…
The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to…
We study clean group rings and also the group rings whose every element is a sum of two units. We also prove that if R is an Abelian exchange ring and G is a locally finite group, then the group ring RG has stable range one.
We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical…
We investigate the dynamical behavior of rings around bodies whose shapes depart considerably from that of a sphere. To this end, we have developed a new self-gravitating discrete element N-body code, and employed a local simulation method…
This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where…
We investigate the least studied class of differential rings -- the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in…