Journal of Algebraic Systems
Volume:10 Issue: 1, Summer-Autumn 2022

We present several results that establish the fusible and the regular left fusible properties of the family of noncommutative rings known as skew Poincar'e-Birkhoff-Witt extensions. Our treatment is based on the recent works of Ghashghaei and McGovern [13], and Kosan and Matczuk [31] concerning the left fusibleness and the regular left fusibleness of skew polynomial rings of automorphism type. Since the results formulated in this paper can be applied to algebraic structures more general than skew polynomial rings, our contribution to the theory of fusibleness is to cover more families of rings of interest in branches as quantum groups, noncommutative algebraic geometry and noncommutative differential geometry. We provide illustrative examples of the ideas developed here.

Further properties on (belligerent) GE-filters are discussed, and the quotient GEalgebra via a GE-filter is established. Conditions for the set →c to be a belligerent GE-filterare provided. The extension property of belligerent GE-filter is composed. The notions of abalanced element, a balanced GE-filter, an antisymmetric GE-algebra and a voluntary GE-filterare introduced, and its properties are examined. The relationship between a GE-subalgebraand a GE-filter is established. Conditions for every element in a GE-algebra to be a balancedelement are provided. The conditions necessary for a GE-filter to be a voluntary GE-filter areconsidered. The GE-filter generated by a given subset is established, and its shape is identified

 This paper extends multirings to a novel concept as general multirings, investigates their properties and presents a special general multirings as notation of (m; n)-potent general multirings. This study analyzes the differences between class of multirings, general multirings and general hyperrings and constructs the class of (in)finite general multirings based on any given non-empty set. In final, we define the concept of hyperideals in general multirings and compare with hyperideals in othersimilar (hyper)structures.

Let a be a proper ideal of a ring R. A finitely generated R-module M is said to be a-relative generalized Cohen-Macaulay if f_a (M)=cd(a ,M). In this note, we introduce a suitable notion of length of a module to characterize the above mentioned modules. Also certain syzygy modules over a relative Cohen-Macaulay ring are studied.

In this paper, the notion of H-sets on Hv-groups is introduced and some related properties are investigated and some examples are given. In this regards, the concept of regular, strongly regular relations and homomorphism of H-sets are adopted. Also, the classical isomorphism theorems of groups are generalized to H-sets on Hv-groups. Finally, by using these concepts tensor product on Hv-groups is introduced andproved that the tensor product exists and is unique up to isomorphism.

Let $G$ be a group with identity $e$, $R$ be an associative graded ring and $M$ be a $G$-graded $R$-module. In this article, we intruduce the concept of graded semiprimesubmodules over non-commutative graded rings. First, we study graded prime $R$-modulesover non-commutative graded rings and we get some properties of such graded modules.Second, we study graded semiprime and graded radical submodules of graded $R$-modules.For example, we give some equivalent conditions for a graded module to have zero gradedradical submodule.

‎For a positive integer $m$‎, ‎a subset of divisors of $m$ is called a textit{divisor topology on $m$} if it contains $1 $ and $m$ and it is closed under taking $gcd$ and $rm lcm$‎. ‎If $m=p_1dots p_n$ is a square free positive integer‎, ‎then a divisor topology $m$ corresponds to a topology on the set $[n]={1,2,ldots,n}$‎. ‎Giving some facts about divisor topologies‎, ‎we give a recursive formula for the number of divisor topologies on a positive integer‎.

In this paper, we construct the concept of general  Krasner  hyperring based on the  ring  structures and the left general Krasner hypermodule based on the  module structures.  This study introduces  the  trivial left general Krasner hypermodules and  proves that the  trivial left general Krasner   hypermodules  are  different from left  Krasner   hypermodules. We show that for any given general  Krasner  hyperring $R$ and trivial left general Krasner   hypermodules $A, B, {bf_{R}h}$om$(A, B)$ is a left general Krasner   hypermodule and  ${bf_{R}h}$om$(-, B)$,     $ ({bf_{R}h}$om$(A, -) )$ is an   exact covariant functor (contravariant). Finally, we  show that the category ${bf_{R}GKH}$mod (left  trivial general Krasner hypermodules and all (homomorphisms) is an abelian  category and  trivial left general Krasner hypermodules have  a normal injective resolution.

A ring R is said to be clean if every element of R is a sumof an idempotent and a unit. A ring R is a neat ring if every nontrivialhomomorphic image is clean. In this paper, first, it is proved that everyelement of some classes of neat rings is n-tuplet-good if no factor ringof such rings isomorphic to a field of order less than n + 2. Also by consideringthe structure of FGC rings, it can be proved that some clasess of FGCrings are n-tuplet-good.

Let G = (V (G); E(G)) be a simple graph. A set D of vertices G is an identifying code of G; if for every two vertices x and y the sets N_G[x] D and N_G[y] D are non- empty and different. The minimum cardinality of an identifying code in graph G is the identifying code number of G and it is denoted by gamma ID(G): Two vertices x and y are twin, when N_G[x] = N_G[y]: Graphs with at least two twin vertices are not identifiable graphs. In this paper, we deal with identifying code number of functigraph of G: Two upper bounds on identifying code number of functigraph are given. Also, we present some graph G with identifying code number |V (G)| - 2.

‎Let $ mathcal{A} $ be a unital algebra over a 2-torsion free commutative ring $ mathcal{R} $ and $ mathcal{M} $ be a unital $ mathcal{A} $-bimodule‎. ‎‎We show taht every Jordan higher derivation $ D={D_n}_{nin mathbb{N}_0} $ from the trivial extension $ mathcal{A} ltimes mathcal{M} $ into itself is a higher derivation, if $ PD_1(QXP)Q=QD_1(PXQ)P=0 $ for all $ X in mathcal{A} ltimes mathcal{M} $‎, in which $ P=(e,0) $ and $ Q=(e^prime,0) $ for some non-trivial idempotent element $ e inmathcal{A} $ and $ e^prime =1_mathcal{A}-e $ satisfying‎‎ ‎the following ‎conditions‎:‎$‎‎emathcal{A}e^primemathcal{A}e={0}‎$‎, ‎$‎e^primemathcal{A}emathcal{A}e^prime={0}‎$‎‎,‎$‎‎e(l.ann_mathcal{A} mathcal{M})e={0}‎$‎‎, ‎$‎e^prime(r.ann_mathcal{A} mathcal{M})e^prime={0}‎$‎‎‎‎‎and $ eme^prime=m $ for all $ m in mathcal{M} $‎.

Let $S_lambda(X)={fin C(X) : |Xsetminus Z(f)|<lambda}$, such that $lambda$ is a regular cardinalnumber with $lambdaleq |X|$.It is generalization of $C_F (X)=S_{aleph_0}(X)$ and$SC_F(X)=S_{aleph_1}(X)$. Usingthis concept we extend some of the basic results concerning the socleto $S_lambda(X)$. It is shown thatif $X$ is a $lambda$-pseudo discrete space, then $C_{K,lambda}(X)subseteq S_{lambda}(X)$.$S_{lambda}$-completely regular spaces are investigated.Consequently, $X$ is a $S_{aleph_1}$-completely regular space if and only if $X$ is $aleph_1$-zero dimensional space.$S_{lambda}P$-spaces are introduced and studied.