Generalized semicommutative and skew Armendariz ideals

We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts.Узагальнено поняття напівкомутативних косих абелевих зведених та симетричних лівих iдеалiв Армендаріза та вивчено співвідношення між ними

Strongly semicommutative rings relative to a monoid

For a monoid M, we introduce strongly M-semicommutative rings obtained as a generalization of strongly semicommutative rings and investigate their properties. We show that if G is a finitely generated Abelian group, then G is torsion free if and only if there exists a ring R with |R| ≥ 2 such that R is strongly G-semicommutative.Для моноїда M, ми вводимо сильно M,-напівкомутативні кільця, що узагальнюють сильно напівкомутативні кільця, та вивчаємо їх властивості. Показано, що якщо G — скінченнопороджена абелева група, то G є вільною від скруту тоді і тільки тоді, коли існує кільце R з |R| ≥ 2 таке, що R є сильно G-напівкомутативним

Properties of a certain product of submodules

Let R be a commutative ring with identity, M an R-module and K1, . . . , Kn submodules of M. In this article, we construct an algebraic object, called product of K1, . . . , Kn. We equipped this structure with appropriate operations to get an R(M)-module. It is shown that R(M)-module Mⁿ = M . . . M and R-module M inherit some of the most important properties of each other. For example, we show that M is a projective (flat) R-module if and only if Mⁿ is a projective (flat) R(M)-moduleПрипустимо, що R — комутативне кiльце з одиницею, M — R-модуль i K1, . . . , Kn — пiдмодулi M. Побудовано алгебраїчний об’єкт, що називається добутком пiдмодулiв K1, . . . , Kn. Цю структуру оснащено вiдповiдними операцiями для отримання R(M)-модуля. Показано, що R(M)-модуль Mⁿ = M . . . M та R-модуль M успадковують деякi з найбiльш важливих властивостей один одного. Наприклад, показано, що M є проективним (плоским) R-модулем тодi i тiльки тодi, коли Mⁿ — проективний (плоский) R(M)-модуль

Weak α-skew Armendariz ideals

We introduce the concept of weak α-skew Armendariz ideals and investigate their properties. Moreover, we prove that I is a weak α-skew Armendariz ideal if and only if I[x] is a weak α-skew Armendariz ideal. As a consequence, we show that R is a weak α-skew Armendariz ring if and only if R[x] is a weak α-skew Armendariz ring.Введено поняття слабких α-косих iдеалiв Армендарiза та дослiджено їх властивостi. Крiм того, доведено, що I є слабким α-косим iдеалом Армендарiза тодi i тiльки тодi, коли I[x] є слабким α-косим iдеалом Армендарiза. Як наслiдок, показано, що R є слабким α-косим кiльцем Армендарiза тодi i тiльки тодi, коли R[x] є слабким α-косим кiльцем Армендарiза

On the inclusion ideal graph of a poset

Let (P,≤) be an atomic partially ordered set (poset, briefly) with a minimum element 0 and \u1d57f(P) the set of nontrivial ideals of P. The inclusion ideal graph of P, denoted by Ω(P), is an undirected and simple graph with the vertex set \u1d57f(P) and two distinct vertices I, J ∈ \u1d57f(P) are adjacent in Ω(P) if and only if I ⊂ J or J ⊂ I. We study some connections between the graph theoretic properties of this graph and some algebraic properties of a poset. We prove that Ω(P) is not connected if and only if P = {0, a1, a2}, where a1, a2 are two atoms. Moreover, it is shown that if Ω(P) is connected, then diam(Ω(P)) ≤ 3. Also, we show that if Ω(P) contains a cycle, then girth(Ω(P)) ∈ {3, 6}. Furthermore, all posets based on their diameters and girths of inclusion ideal graphs are characterized. Among other results, all posets whose inclusion ideal graphs are path, cycle and star are characterized

On the coloring of the annihilating-ideal graph of a commutative ring

AbstractSuppose that R is a commutative ring with identity. Let A(R) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)∗=A(R)∖{(0)} and two distinct vertices I and J are adjacent if and only if IJ=(0). In Behboodi and Rakeei (2011) [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, girth(AG(R))=3. Here, we prove that for every (not necessarily reduced) ring R, ω(AG(R))≥|Min(R)|, which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. Among other results, it is shown that if the chromatic number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite too. We investigate commutative rings whose annihilating-ideal graphs are bipartite. It is proved that AG(R) is bipartite if and only if AG(R) is triangle-free