Journal of Algebraic Systems
Volume:10 Issue: 2, Winter-Spring 2023

‎A perfect Roman dominating function (PRDF) on a graph $G$ is a function $ f:V(G)to {0,1,2}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$‎. ‎The weight of a PRDF $f$ is the sum of the weights of the vertices under $f$‎. ‎The perfect Roman domination number of $G$ is the minimum weight of a PRDF in $G$‎. ‎In this paper we study algorithmic and computational complexity aspects of the minimum perfect Roman domination problem (MPRDP)‎. ‎We first correct the proof of a result published in [Bulletin‎‎Iran‎. ‎Math‎. ‎Soc‎. ‎14(2020)‎, ‎342--351]‎, ‎and using a similar argument‎, ‎show that MPRDP is APX-hard for graphs with bounded degree 4‎.‎We prove that the decision problem associated to MPRDP is NP-complete even when restricted to star convex bipartite graphs‎. ‎Moreover‎, ‎we show that MPRDP is solvable in linear time for bounded tree-width‎‎graphs‎. ‎We also show that the perfect domination problem and perfect Roman domination problem are not equivalent in computational complexity aspects‎. ‎Finally we propose an integer linear programming formulation for MPRDP‎.

An associative ring R with identity is called r¡clean ring if everyelement of R is the sum of a regular and an idempotent element. In this paper,we introduce the concept of r-clean rings relative to right ideal. We studyvarious properties of these rings. We give some relations between r-cleanrings and r-clean rings of 2 2 matrices over R relative to some right idealP. New characterization obtained include necessary and sufficient conditionsof a ring R to be r-clean in terms of P-regular, P-local and P-clean rings.Also, We prove that every ring is r-clean relative to any maximal right idealof it.

Let $R= bigoplus_{g in G} R_g$ be a $G-$graded commutative ring with identity, $I$ be a graded ideal and let $M$ a $G-$graded unitary $R$-module, where $G$ is a semigroup with identity $e$. We introduce graded $I-$prime ideals (submodules) as a generalizations of the classical notions of prime ideals (submodules). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module $F$, a graded submodule $P$ of $M$ is $I-$prime if and only if $F otimes P$ is graded $I-$prime submodule of $F otimes M$. As an application, for finitely generated graded module $M$ over Noetherian graded ring $R$, the completion of graded $I-$prime submodules is $I-$prime submodule.

Let R be a commutative Noetherian ring with nonzero identity. Let φ be a system of ideals of R and let M, N two finitely generated R-modules. We prove that there are local- global principles for the finiteness and minimaxness of generalized local cohomology module H_φ^i (M, N) , in certain cases.

In this paper, we introduce Condition (PWPsc) as a generalization of Condition (PWP_E) of acts over monoids, and we observe that Condition (PWPsc) does not imply Condition (PWP_E). In general, we show that Condition (PWPsc) implies the property of being principally weakly flat, and that in left PSFmonoids, the converse of this implication is also true. Moreover, we present some general properties and a homological classification of monoids by comparing Condition (PWPsc) with some other properties. Finally, we describe left PSF monoids for which S^I_S satisfies Condition (PWPsc) for any nonempty set I.

The notion of commutative falling intuitionistic fuzzy ideal of a BCK-algebra is introduced and related properties are investigated. We verify that every commutative intuitionistic fuzzy ideal is a commutative falling intuitionistic fuzzy ideal, and provide example to show that a commutative falling intuitionistic fuzzy ideal is not a commutative intuitionistic fuzzy ideal. Relations between a falling intuitionistic fuzzy ideal and a commutative falling intuitionistic fuzzy ideal are considered, and a condition for a falling intuitionistic fuzzy ideal to be a commutative falling intuitionistic fuzzy ideal is provided.

The distance eigenvalues of a connected graph $G$ are the eigenvalues of its distance matrix‎‎$D(G)$‎. ‎A graph is called distance integral if all of its‎‎distance eigenvalues are integers.‎‎Let $n$ and $k$ be integers with $n>2k‎, ‎kgeq1$‎. ‎The bipartite Kneser graph $H(n,k)$ is the graph with the set of all $k$ and $n-k$ subsets of the set $[n]={1,2,...,n}$ as vertices‎, ‎in which two vertices are adjacent if and only if one of them is a subset of the other‎. ‎In this paper‎, ‎we determine the distance spectrum of $H(n,1)$‎. ‎Although the obtained result is not new cite{12}‎, ‎but our proof is new‎. ‎The main tool that we use in our work is the orbit partition method in algebraic graph theory for finding the eigenvalues of graphs‎. ‎We introduce a new method for‎‎determining the distance spectrum of $H(n,1)$ and show how‎‎a quotient matrix can contain all distance eigenvalues of‎‎a graph.‎

In this paper, we introduce a path hyperoperation associated with a hypergraph,which is an extension of the Corsini’s hyperoperation.We investigate some related properties and study relations betweenthe path hyperoperation and hypergraph theory.

Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionallycountable subalgebra of $C(X)$ (resp., $C^*(X)$),consisting of all functions (resp., bounded functions) with countable image.$C_c(X)$ (resp., $C_c^*(X)$) as a topological ring via $m_c$-topology (resp., $m^*_c$-topology) and $u_c$-topology (resp., $u^*_c$-topology) is investigated and the equality of the latter two topologies is characterized. Topological spaces which are called $N$-spaces are introduced and studied.It is shown that the $m_c$-topology on $C_c(X)$ and its relative topology as a subspace of $C(X)$ (with $m$-topology) coincide if and only if $X$ is an $N$-space. We also show that $X$ is pseudocompact if and only if it is both a countably pseudocompact, and an $N$-space.

‎Let $R$ be a commutative ring and $Z(R)$ be the set of its zero-divisors‎.‎The annihilator graph of $R$‎, ‎denoted by $AG(R)$ is a simple undirected graph whose vertex‎‎set is $Z(R)^*$‎, ‎the set of all nonzero zero-divisors of $R$‎, ‎and two distinct vertices $x$ and‎‎$y$ are adjacent if and only if ${rm ann}_R(xy)neq {rm ann}_R(x)cup {rm ann}_R(y)$‎.‎In this paper‎, ‎perfectness of the annihilator graph for some classes of rings is investigated‎.‎More precisely‎, ‎we show that if $R$ is an Artinian ring‎, ‎then $AG(R)$ is perfect‎.

Let $L$ be a lattice with the least element $0$ and the greatest element $1$. In this paper, we associate a graph to filters of $L$, in which the vertex set is being the set of all non-trivial filters of $L$, and two distinct vertices $F$ and $E$ are adjacent if and only if $F cap E neq {1}$. We denote this graph by $mathcal{G}$ $(L)$. The basic properties and possible structures of $mathcal{G}$ $(L)$ are studied. Moreover, we characterize the planarity of $mathcal{G}$ $(L)$.

We say a ring R with unity is left weakly Baer if the left annihilatorof any nonempty subset of R is right s-unital by right semicentral idempotents,which implies that R modulo the left annihilator of any nonempty subset isflat. It is shown that, unlike the Baer or right PP conditions, the weaklyBaer property is inherited by polynomial extensions. Examples are providedto explain the results.