Papers by Farid Aliniaeifard
Let $S$ be a semigroup with $0$ and $R$ be a ring with $1$. We extend the definition of the zero-... more Let $S$ be a semigroup with $0$ and $R$ be a ring with $1$. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}(S)$, and the other definition yields an undirected graph $\overline{{\Gamma}}(S)$. It is shown that $\Gamma(S)$ is not necessarily connected, but $\overline{{\Gamma}}(S)$ is always connected and ${\rm diam}(\overline{\Gamma}(S))\leq 3$. For a ring $R$ define a directed graph $\Bbb{APOG}(R)$ to be equal to $\Gamma(\Bbb{IPO}(R))$, where $\Bbb{IPO}(R)$ is a semigroup consisting of all products of two one-sided ideals of $R$, and define an undirected graph $\overline{\Bbb{APOG}}(R)$ to be equal to $\overline{\Gamma}(\Bbb{IPO}(R))$. We show that $R$ is an Artinian (resp., Noetherian) r...
Let $R$ be a commutative ring with $1\neq 0$ and $\Bbb{A}(R)$ be the set of ideals with nonzero a... more Let $R$ be a commutative ring with $1\neq 0$ and $\Bbb{A}(R)$ be the set of ideals with nonzero annihilators. The annihilating-ideal graph of $R$ is defined as the graph $\Bbb{AG}(R)$ with the vertex set $\Bbb{A}(R)^{*} = \Bbb{A}(R)\setminus \{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ = (0)$. In this paper, we first study the interplay between the diameter of annihilating-ideal graphs and zero-divisor graphs. Also, we characterize rings $R$ when ${\rm gr}(\Bbb{AG}(R))\geq 4$, and so we characterize rings whose annihilating-ideal graphs are bipartite. Finally, in the last section we discuss on a relation between the Smarandache vertices and diameter of $\Bbb {AG}(R)$.
Communications in Algebra, 2014
Let R be a commutative ring with 1 = 0, G be a nontrivial finite group and let Z(R) be the set of... more Let R be a commutative ring with 1 = 0, G be a nontrivial finite group and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R) * = Z(R) \ {0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤ 2 or gr(Γ(RG)) ≥ 4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.
Journal of Pure and Applied Algebra, 2013
A group G is called morphic if every endomorphism : G ! G for which G is normal in G satis…es G=G... more A group G is called morphic if every endomorphism : G ! G for which G is normal in G satis…es G=G = ker( ): This concept for modules was …rst investigated by G. Ehrlich in 1976. Since then the concept has been extensively studied in module and ring theory. A recent paper of Li, Nicholson and Zan investigated the idea in the category of groups. A characterization for a …nite nilpotent group to be morphic was obtained, and some results about when a small p-group is morphic were given. In this paper, we continue the investigation of the general …nite morphic p-groups. Necessary and su¢ cient conditions for a morphic p-group of order p n (n > 3) to be abelian are given. Our main results show that if G is a morphic p-group of order p n with n > 3 such that either d(G) = 2 or jG 0 j < p 3 , then G is abelian, where d(G) is the minimal number of generators of G. As consequences of our main results we show that any morphic p-groups of order p 4 ; p 5 and p 6 are abelian.
Communications in Algebra, 2013
It was shown by C. Wickham [12] that "for a fixed positive integer g, there are finitely many iso... more It was shown by C. Wickham [12] that "for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g". In this note we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.
Communications in Algebra, 2014
ABSTRACT For a commutative ring R with identity, the annihilating-ideal graph of R, denoted (R), ... more ABSTRACT For a commutative ring R with identity, the annihilating-ideal graph of R, denoted (R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted I (R). We discuss when I (R) is bipartite. We also give some results on the subgraphs and the parameters of I (R).
Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators.... more Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We investigate commutative rings $R$ whose annihilating-ideal graphs have positive genus $\gamma(\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $\gamma(\Bbb{AG}(R))<\infty$, then $R$ has finitely many ideals or $(R,\mathfrak{m})$ is a Gorenstein ring with maximal ideal $\mathfrak{m}$ and ${\rm v.dim}_{R/{\mathfrak{m}}}{\mathfrak{m}}/{\mathfrak{m}}^{2}=2$. Also, for any two integers $g\geq 0$ and $q>0$, there are finitely many isomorphism classes of Artinian rings $R$ satisfying the conditions: (i) $\gamma(\Bbb{AG}(R)) < g$ and (ii) $|R/{\mathfrak{m}}| \leq q$ for every maximal ideal ${\mathfrak{m}}$ of $R$. Also, it is shown that if $R$ is a non-domain Noetherian local ring such that $\gamma(\Bbb{AG}(R))<\infty$, then either $R$ is a Gorenstein ring or $R$ is an Artinian ring with finitely many ideals.
Let R be a commutative ring with 1 and let I(R) be the set of all proper ideals
of R. An ideal I... more Let R be a commutative ring with 1 and let I(R) be the set of all proper ideals
of R. An ideal I in I(R) is called an annihilator ideal of R if, IJ = 0 for some nonzero
ideal J of I(R). Let A(R) denote the set of all annihilators ideals of R. In this paper, we
define the Annihilating-Ideal graph of R (denoted by AG(R)), as an undirected graph
with vertices A(R)* = A(R)\ {(0)}, where distinct vertices I and J are adjacent if and
only if IJ = 0. We investigate commutative rings whose annihilating-ideal graphs have
positive genus. It is shown that if R is an Artinian ring then, 0 < g(AG(R)) < 1 if
and only if R has finitely many ideals. Also prove for two integer g>0 and q > 0,
there are finitely many Artinian rings R such that they satisfy in following conditions
(1) g(AG(R)) < g (2) |R/m| < q+1 for every maximal ideal m of R. The proves can be
modified to yield an analogous result for nonorientable genus.
Key Words: Commutative ring; Annihilating-Ideal graph; genus of a Graph
For a commutative ring R with identity, the annihilatingideal
graph of R, denoted AG(R), is the ... more For a commutative ring R with identity, the annihilatingideal
graph of R, denoted AG(R), is the graph whose vertices are
the nonzero annihilating ideal of R with two distinct vertices joined
by an edge when the product of the vertices is zero. This article
extend the definition of the annihilating ideal graph to noncommutative
rings.
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Papers by Farid Aliniaeifard
of R. An ideal I in I(R) is called an annihilator ideal of R if, IJ = 0 for some nonzero
ideal J of I(R). Let A(R) denote the set of all annihilators ideals of R. In this paper, we
define the Annihilating-Ideal graph of R (denoted by AG(R)), as an undirected graph
with vertices A(R)* = A(R)\ {(0)}, where distinct vertices I and J are adjacent if and
only if IJ = 0. We investigate commutative rings whose annihilating-ideal graphs have
positive genus. It is shown that if R is an Artinian ring then, 0 < g(AG(R)) < 1 if
and only if R has finitely many ideals. Also prove for two integer g>0 and q > 0,
there are finitely many Artinian rings R such that they satisfy in following conditions
(1) g(AG(R)) < g (2) |R/m| < q+1 for every maximal ideal m of R. The proves can be
modified to yield an analogous result for nonorientable genus.
Key Words: Commutative ring; Annihilating-Ideal graph; genus of a Graph
graph of R, denoted AG(R), is the graph whose vertices are
the nonzero annihilating ideal of R with two distinct vertices joined
by an edge when the product of the vertices is zero. This article
extend the definition of the annihilating ideal graph to noncommutative
rings.
of R. An ideal I in I(R) is called an annihilator ideal of R if, IJ = 0 for some nonzero
ideal J of I(R). Let A(R) denote the set of all annihilators ideals of R. In this paper, we
define the Annihilating-Ideal graph of R (denoted by AG(R)), as an undirected graph
with vertices A(R)* = A(R)\ {(0)}, where distinct vertices I and J are adjacent if and
only if IJ = 0. We investigate commutative rings whose annihilating-ideal graphs have
positive genus. It is shown that if R is an Artinian ring then, 0 < g(AG(R)) < 1 if
and only if R has finitely many ideals. Also prove for two integer g>0 and q > 0,
there are finitely many Artinian rings R such that they satisfy in following conditions
(1) g(AG(R)) < g (2) |R/m| < q+1 for every maximal ideal m of R. The proves can be
modified to yield an analogous result for nonorientable genus.
Key Words: Commutative ring; Annihilating-Ideal graph; genus of a Graph
graph of R, denoted AG(R), is the graph whose vertices are
the nonzero annihilating ideal of R with two distinct vertices joined
by an edge when the product of the vertices is zero. This article
extend the definition of the annihilating ideal graph to noncommutative
rings.