# Showing papers in "Journal of Algebra and Related Topics in 2015"

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TL;DR: In this paper, a generalization of the small submodules, called small sub-modules, is introduced, where the sub-module is called small in a ring and the ring is an arbitrary ring.

Abstract: Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$-module $M$. A submodule $N$ of $M$ is called $T$-small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

8 citations

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TL;DR: In this paper, the Eulerian properties of the line graph associated to the maximal graph of a commutative ring with identity were studied. And the authors showed that the complement of a maximal graph is a Euler graph if and only if the graph has odd cardinality.

Abstract: Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphical properties of the line graph associated to $Gamma(R)$, denoted by $(Gamma(R))$ such that diameter, completeness, and Eulerian property. A complete characterization of rings is given for which $diam(L(Gamma(R)))= diam(Gamma(R))$ or $diam(L(Gamma(R))) diam(Gamma(R))$. We have shown that the complement of the maximal graph $G(R)$, i.e., the comaximal graph is a Euler graph if and only if $R$ has odd cardinality. We also discuss the Eulerian property of the line graph associated to the comaximal graph.

4 citations

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TL;DR: In this article, the dual notion of strongly top modules is introduced and the basic properties of this class of modules are studied, as well as some properties of these classes of modules.

Abstract: In this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.

3 citations

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TL;DR: In this paper, the authors assert some properties of finitely generated comultiplication modules and fit ideals of them, and show that these properties can be found in a commutative ring.

Abstract: Let $R$ be a commutative ring. In this paper we assert some properties of finitely generated comultiplication modules and Fitting ideals of them.

1 citations

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TL;DR: In this paper, the authors investigate the total graph of a commutative semiring with respect to a given semiring, denoted by the subgraphs of the complete graph of the semiring.

Abstract: Let $I$ be a proper ideal of a commutative semiring $R$ and let $P(I)$ be the set of all elements of $R$ that are not prime to $I$. In this paper, we investigate the total graph of $R$ with respect to $I$, denoted by $T(Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y in P(I)$. The properties and possible structures of the two (induced) subgraphs $P(Gamma_{I} (R))$ and $bar {P}(Gamma_{I} (R))$ of $T(Gamma_{I} (R))$, with vertices $P(I)$ and $R - P(I)$, respectively are studied.

1 citations

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TL;DR: In this article, the authors define the notion of maximal non-maximal ideal of a ring and a ring's proper ideal, i.e., a ring ideal that is maximal with respect to the property of not being a prime ideal.

Abstract: The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $Ineq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$, $I$ is maximal with respect to the property of being not a prime ideal. The concept of maximal non-maximal ideal and maximal non-primary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals $I$ of a ring $R$ such that $I$ is a maximal non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of $R$.

1 citations

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TL;DR: Weakly $J$-quasipolar rings were introduced in this article, where a ring $R$ is called weakly quasipolar if and only if it is uniquely clean.

Abstract: In this paper, we introduce a class of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {it weakly $J$-quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $a-p$ are contained in $J(R)$ and the ring $R$ is called {it weakly $J$-quasipolar} if every element of $R$ is weakly $J$-quasipolar. We give many characterizations and investigate general properties of weakly $J$-quasipolar rings. If $R$ is a weakly $J$-quasipolar ring, then we show that (1) $R/J(R)$ is weakly $J$-quasipolar, (2) $R/J(R)$ is commutative, (3) $R/J(R)$ is reduced. We use weakly $J$-quasipolar rings to obtain more results for $J$-quasipolar rings. We prove that the class of weakly $J$-quasipolar rings lies between the class of $J$-quasipolar rings and the class of quasipolar rings. Among others it is shown that a ring $R$ is abelian weakly $J$-quasipolar if and only if $R$ is uniquely clean.

1 citations

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TL;DR: In this paper, it was shown that the polynomial ring over a field of positive integers can be shown to have three monomial complete intersections if I, J, K subseteq R$ are three complete intersections.

Abstract: Let $R=k[x_1,x_2,cdots, x_N]$ be a polynomial ring over a field $k$. We prove that for any positive integers $m, n$, $text{reg}(I^mJ^nK)leq mtext{reg}(I)+ntext{reg}(J)+text{reg}(K)$ if $I, J, Ksubseteq R$ are three monomial complete intersections ($I$, $J$, $K$ are not necessarily proper ideals of the polynomial ring $R$), and $I, J$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l}^{a_l})$.

1 citations

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TL;DR: In this paper, the authors generalize some of known results on tight closure of an ideal to the tight closure relative to a module, and show that the latter is tight relative to the latter.

Abstract: In this paper we will generalize some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

1 citations

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TL;DR: In this article, the authors considered the finitely presented groups $G_{m}$ and $K(s,l)$ and found the $n^{th}$-commutativity degree for each of them.

Abstract: In this paper, we consider the finitely presented groups $G_{m}$ and $K(s,l)$ as follows;$$G_{m}=langle a,b| a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]rangle $$$$K(s,l)=langle a,b|ab^s=b^la,~ba^s=a^lbrangle;$$and find the $n^{th}$-commutativity degree for each of them. Also we study the concept of $n$-abelianity on these groups, where $m,n,s$ and $l$ are positive integers, $m,ngeq 2$ and $g.c.d(s,l)=1$.

1 citations

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TL;DR: In this paper, the authors consider all finite non-abelian 2-generator groups of nilpotency class two and study the probability of having $n −th −1 root in these groups.

Abstract: Here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. Also we find integers $n$ for which, these groups are $n$-central.