arXiv subjects

Mahmood Behboodi

Publications and source records attributed to Mahmood Behboodi.

16 recordsLinked to original sources

arXiv2023

Left Co-Köthe Rings and Their Characterizations

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings $R$ such that every left $R$-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic fnite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper ["Several characterizations of left Köthe rings", submitted], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe ring, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them.

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arXiv2022

Several Characterizations of Left Köthe Rings

We study the classical Köthe's problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring $R$ is called a ${\it left~Köthe~ring}$ if every left $R$-module is a direct sum of cyclic $R$-modules. In 1951, Cohen and Kaplansky proved that all commutative K{ö}the rings are Artinian principal ideal rings. During the years 1962 to 1965, Kawada solved the Köthe's problem for basic fnite-dimensional algebras: Kawada's theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has square-free socle and square-free top, and describes the possible indecomposable modules. But, so far, the Köthe's problem is open in the non-commutative setting. In this paper, we break the class of left K{ö}the rings into three categories of nested: ${\it left~Köthe~rings}$, ${\it strongly~left~K{ö}the~rings}$ and ${\it very~strongly~left~K{ö}the~rings}$, and then, we solve the Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of Köthe-Cohen-Kaplansky theorem.

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arXiv2022

$Σ$-semi-compact rings and modules

In this paper several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property $Σ$-semi-compact for modules and we characterize the modules satisfying this property. In particular, we show that a ring $R$ is left $Σ$-semi-compact if and only if $R$ satisfies the ascending (resp. descending) chain condition on the left (resp. right) annulets. Moreover, we prove that every flat left $R$-module is semi-compact if and only if $R$ is left $Σ$-semi-compact. We also show that a ring $R$ is left Noetherian if and only if every pure projective left $R$-module is semi-compact. Finally, we consider rings whose flat modules are finitely (singly) projective. For any commutative arithmetical ring $R$ with quotient ring $Q$, we prove that every flat $R$-module is semi-compact if and only if every flat $R$-module is finitely (singly) projective if and only if $Q$ is pure semisimple. A similar result is obtained for reduced commutative rings $R$ with the space $\mathrm{Min}\ R$ compact. We also prove that every $(\aleph_{0},1)$-flat left $R$-module is singly projective if $R$ is left $Σ$-semi-compact, and the converse holds if $R^{\mathbb{N}}$ is an $(\aleph_{0},1)$-flat left $R$-module.

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arXiv2016

Two Generalizations of the Wedderburn-Artin Theorem with Applications

We say that an $R$-module $M$ is {\it virtually simple} if $M\neq (0)$ and $N\cong M$ for every non-zero submodule $N$ of $M$, and {\it virtually semisimple} if each submodule of $M$ is isomorphic to a direct summand of $M$. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules. Our theory provides two natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring $R$: (i) Every finitely generated left (right) $R$-modules is virtually semisimple; (ii) Every finitely generated left (right) $R$-modules is a direct sum of virtually simple modules; (iii) $R\cong\prod_{i=1}^{k} M_{n_i}(D_i)$ where $k, n_1,\ldots,n_k\in \Bbb{N}$ and each $D_i$ is a principal ideal V-domain; and {\rm (iv)} Every non-zero finitely generated left $R$-module can be written uniquely (up to isomorphism and order of the factors) in the form $ Rm_1 \oplus\ldots\oplus Rm_k$ where each $Rm_i$ is either a simple $R$-module or a left virtually simple direct summand of $R$.

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arXiv2016

Virtually Semisimple Modules and a Generalization of the Wedderburn-Artin Theorem

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them {\it virtually semisimple modules}. A module $_RM$ is called {\it completely virtually semisimple} if each submodules of $M$ is a virtually semisimple module. A ring $R$ is then called {\it left} ({\it completely}) {\it virtually semisimple} if $_RR$ is a left (compleatly) virtually semisimple $R$-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring $R$ is left completely virtually semisimple if and only if $R \cong \prod _{i=1}^ k M_{n_i}(D_i)$ where $k, n_1, ...,n_k\in \Bbb{N}$ and each $D_i$ is a principal left ideal domain. Moreover, the integers $k,~ n_1, ...,n_k$ and the principal left ideal domains $D_1, ...,D_k$ are uniquely determined (up to isomorphism) by $R$.

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arXiv2015

Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

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arXiv2013

Commutative Local Rings whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.

math.AC↗

arXiv2012

On rings each of whose finitely generated modules is a direct sum of cyclic modules

In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by various authors. It is shown that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. In particular, it is shown that a Noetherian local duo-ring $R$ is a left FGC-ring if and only if $R$ is a right FGC-ring, if and only if, $R$ is a principal ideal ring. Moreover, we obtain that if $R=Π_{i=1}^n R_i$ is a finite product of Noetherian duo-rings $R_i$ where each $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ is a principal ideal ring.each $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ is a principal ideal ring.

math.RA↗

arXiv2012

On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \cal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\cal{M}}=\bigoplus_{λ\in Λ}Rw_λ$ and $R/{\rm Ann}(w_λ)$ is a principal ideal ring for each $λ\in Λ$;(3) Every prime ideal of $R$ is a direct sum of at most $|Λ|$ cyclic $R$-modules; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \cal{M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\cal{M}$.

math.AC↗

arXiv2012

Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules I

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{\cal{P}M}:M)={\cal{P}}$" for every prime ideal ${\cal{P}}\supseteq {\rm Ann}(M)$, where $\sqrt[p]{\cal{P}M}$ is the intersection of all prime submodules of $M$ containing ${\cal{P}}M$). This class contains the family of primeful modules properly. This yields that over any ring all free modules and all finitely generated modules lie in the class of $\mathbb{P}$-radical modules. Also, we show that if $R$ is a domain (or a Noetherian ring), then all projective modules are $\mathbb{P}$-radical. In particular, if $R$ is an Artinian ring, then all $R$-modules are $\mathbb{P}$-radical and the converse is also true when $R$ is a Noetherian ring. Also an $R$-module $M$ is called $\mathbb{M}$-radical if $(\sqrt[p]{\cal{M}M}:M)={\cal{M}}$; for every maximal ideal ${\cal{M}}\supseteq {\rm Ann}(M)$. We show that the two concepts $\mathbb{P}$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Semisimple $\mathbb{P}$-radical ($\mathbb{M}$-radical) modules are also characterized. In Part II we shall continue the study of this construction, and as an application, we show that the sheaf theory of spectrum of $\mathbb{P}$-radical modules (with the Zariski topology) resembles to that of rings.

math.AC↗

arXiv2012

Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules II

In this paper we continue our study of modules satisfying the prime radical condition ($\mathbb{P}$-radical modules), that was introduced in Part I (see \cite{BS}). Let $R$ be a commutative ring with identity. The purpose of this paper is to show that the theory of spectrum of $\mathbb{P}$-radical $R$-modules (with the Zariski topology) resembles to that of rings. First, we investigate the behavior of $\mathbb{P}$-radical modules under localization and direct sums. Finally, we describe the construction of a structure sheaf on the prime spectrum Spec$(M)$, which generalizes the classical structure sheaf of the ring $R$ in Algebraic Geometry to the module $M$.

math.AC↗

arXiv2012

Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that {\it classical prime} submodules are the intersection of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings $R$ over which all $R$-modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings $R$ for which all finitely generated $R$-modules are classical Hilbert.

math.AC↗

arXiv2012

Prime M-Ideals, M-Prime Submodules, M-Prime Radical and M-Baer's Lower Nilradical of Modules

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-m-system sets, M-n-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "prime M-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfes condition H (defined latter) and Hom_R(M,X)\neq 0$ for all modules X in the category σ[M], then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in σ[M] and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.

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arXiv2011

The Annihilating-Ideal Graph of Commutative Rings I

Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is the (undirected) graph with vertices ${\Bbb{A}}(R)^*:={\Bbb{A}}(R)\setminus\{(0)\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness conditions of ${\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a domain, then ${\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if $R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of ${\Bbb{AG}}(R)$ and the set of nonzero proper ideals of $R$ have the same cardinality when $R$ is either an Artinian or a decomposable ring. This yields for a ring $R$, ${\Bbb{AG}}(R)$ has $n$ vertices $(n\geq 1)$ if and only if $R$ has only $n$ nonzero proper ideals. Next, we study the connectivity of ${\Bbb{AG}}(R)$. It is shown that ${\Bbb{AG}}(R)$ is a connected graph and $diam(\Bbb{AG})(R)\leq 3$ and if ${\Bbb{AG}}(R)$ contains a cycle, then $gr({\Bbb{AG}}(R))\leq 4$. Also, rings $R$ for which the graph ${\Bbb{AG}}(R)$ is complete or star, are characterized, as well as rings $R$ for which every vertex of ${\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

math.AC↗

arXiv2011

The Annihilating-Ideal Graph of Commutative Rings II

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [5]). Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator and $Z(R)$ its set of zero divisors. The annihilating-ideal graph of $R$ is defined as the (undirected) graph ${\Bbb{AG}}(R)$ that its vertices are $\Bbb{A}(R)^* =\Bbb{A}(R)\hspace{-1mm}\setminus\{(0)\}$ in which for every distinct vertices $I$ and $J$, $I\hspace{-0.6mm}-\hspace{-1.7mm}-\hspace{-1.7mm}-\hspace{-0.5mm}J$ is an edge if and only if $IJ=(0)$. First, we study the diameter of ${\Bbb{AG}}(R)$. A complete characterization for the possible diameter is given exclusively in terms of the ideals of $R$ when either $R$ is a Noetherian ring or $Z(R)$ is not an ideal of $R$. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either $χ({\Bbb{AG}}(R))\leq 2$ or $R$ is reduced and $χ({\Bbb{AG}}(R))\leq \infty$. Also it is shown that for each reduced ring $R$, $χ(\Bbb{AG}(R))= cl(\Bbb{AG}(R))$. Moreover, if $χ(\Bbb{AG}(R))$ is finite, then $R$ has a finite number of minimal primes, and if $n$ is this number, then $χ(\Bbb{AG}(R))= cl(\Bbb{AG}(R))= n$. Finally, we show that for a Noetherian ring $R$, $cl(\Bbb{AG}(R))$ is finite if and only if for every ideal $I$ of $R$ with $I^2=(0)$, $I$ has finite number of $R$-submodules.

math.AC↗

arXiv2011

Rings Whose Annihilating-Ideal Graphs Have Positive Genus

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 $γ(\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $γ(\Bbb{AG}(R))<\infty$, 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) $γ(\Bbb{AG}(R)) < g$ 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 $γ(\Bbb{AG}(R))<\infty$, then either $R$ is a Gorenstein ring or $R$ is an Artinian ring with finitely many ideals.

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