Proper Submodule Research Articles

On polynomial representations of strange Lie superalgebras of Q-type

In this paper, both canonical and noncanonical polynomial representations of Lie superalgebara of Q-type are investigated. It turns out that not all these representations are completely reducible. Moreover, the representation spaces has only two proper submodules when it is completely reducible, and has a unique composition series when it is not completely reducible.

On 2-Absorbing Primary Submodules of Modules over Commutative Rings

Abstract All rings are commutative with 1 ≠ 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a; b ∈ R and m ∈ M and abm ∈ N, then am ∈ M-rad(N) or bm ∈ M-rad(N) or ab ∈(N :R M). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K ⊆ N for some ideals I1; I2 of R and some submodule K of M, then I1I2 ⊆ (N :R M) or I1K ⊆ M-rad(N) or I2K ⊆ M-rad(N). We prove that for a submodule N of an R-module M if M-rad(N) is a prime submodule of M, then N is a 2-absorbing primary submodule of M. If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M, then (N :R M) is a 2-absorbing primary ideal of R and M-rad(N) is a 2-absorbing submodule of M.

Weakly $2$-absorbing submodules of modules

Let $M$ be a module over a commutative ring $R.$ A proper submodule $N$ of $M$ is called weakly $2$-absorbing, if for $r,s\in R$ and $x\in M$ with $0\neq rsx\in N,$ either $rs\in (N:M)$ or $rx\in N$ or $sx\in N.$ We study the behavior of $(N:M)$ and $\sqrt{(N:M)},$ when $N$ is weakly $2$-absorbing. The weakly $2$-absorbing submodules when $R=R_1\oplus R_2$ are characterized. Moreover we characterize the faithful modules whose proper submodules are all weakly $2$-absorbing.

On weakly classical primary submodules

In this paper all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a {\it classical primary submodule}, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that either $am\in N$ or $b^tm\in N$ for some $t\geq1$. We introduce the notion of ``weakly classical primary submodules''. A proper submodule $N$ of $M$ is a {\it weakly classical primary submodule} if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then either $am\in N$ or $b^tm\in N$ for some $t\geq1$.

On 2-absorbing primary submodules of modules over commutative ring with unity

In this paper, we introduce the concept of a [Formula: see text]-absorbing primary submodule over a commutative ring with nonzero identity which is a generalization of primary submodule. Let [Formula: see text] be an [Formula: see text]-module and [Formula: see text] be a proper submodule of [Formula: see text]. Then [Formula: see text] is said to be a [Formula: see text]-absorbing primary submodule of [Formula: see text] if whenever [Formula: see text], [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. We have given an example and proved number of results concerning [Formula: see text]-absorbing primary submodules. We have also proved the [Formula: see text]-absorbing primary avoidance theorem for submodules.

The Nakayama Property of a Module and Related Concepts

Three related properties of a module are investigated in this article, namely the Nakayama property, the Maximal property, and the S-property. A module M has the Nakayama property if 𝔞M = M for an ideal 𝔞 implies that sM = 0 for some s ∈ 𝔞 + 1. A module M has the Maximal property if there is in M a maximal proper submodule, and finally, M is said to have the S-property if S −1 M = 0 for a multiplicatively closed set S implies that sM = 0 for some s ∈ S.

Minimal faithful modules over Artinian rings

Let R be a left Artinian ring, and M a faithful left R-module which is minimal, in the sense that no proper submodule or proper homomorphic image of M is faithful. If R is local, and socle(R) is central in R, we show that length(M/J(R)M)+length(socle(M))$\leq$ length(socle(R))+1, strengthening a result of T. Gulliksen. Most of the rest of the paper is devoted to the situation where the Artinian ring R is not necessarily local, and does not necessarily have central socle. In the case where R is a finite-dimensional algebra over an algebraically closed field, we get an inequality similar to the above, with the length of socle(R) interpreted as its length as a bimodule, and with the final summand +1 replaced by the Euler characteristic of a bipartite graph determined by the structure of socle(R). More generally, that inequality holds if, rather than assuming k algebraically closed, we assume that R/J(R) is a direct product of full matrix algebras over k, and exclude the case where k has small finite cardinality. Examples show that the restriction on the cardinality of k is needed; we do not know whether versions of our result hold with other hypotheses significantly weakened. The situation for faithful modules with only one minimality property, i.e., having no faithful proper submodules or having no faithful proper homomorphic images, is more straightforward: The length of M/J(R)M in the former case, and of socle(M) in the latter, is $\leq$ length(socle(R)) (again meaning length as a bimodule). The final section, essentially independent of the rest of the note, obtains these bounds, and shows that every faithful module over a left Artinian ring has a faithful submodule with the former minimality condition and a faithful factor module with the latter. The proofs involve some nice general results on decompositions of modules.

S-maximal Submodules

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ? W ? M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of rings and modules. .

S-maximal Submodules

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ? W ? M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of rings and modules. .

On Co-Primarily Packed Modules

Let R be a commutative ring with 1, and M is a (left) R-module. We introduce the concept of coprimarily packed submodules as a proper submodule N of an R-module M which is said to be Coprimarily Packed Submodule. If where Na is a primary submodule of M for each , then for some . When there exists such that ; N is called Strongly Coprimarily Packed submodule. In this paper, we list some basic properties of this concept. We end this paper by explaining the relations between p-compactly packed and coprimarily packed submodules, and also the relations between strongly p-compactly packed and strongly coprimarily packed submodules.

QUASI-PRIMARY SUBMODULES SATISFYING THE PRIMEFUL PROPERTY II

Let $R$ be a commutative ring with identity and $M$ a unital $R$-module. In this article we extend the notion of quasi-primary ideals to submodules. A proper submodule $N$ of $M$ is called quasi-primary if whenever $rx\in N$ for $r\in R$ and $x\in M$, then $r\in \sqrt{(N:M)}$ or $x\in radN$ where $radN$ is the intersection of all prime submodules of $M$ containing $N$. Also, we say that a submodule $N$ of $M$ satisfies the primeful property if $M/N$ is a primeful $R$-module. For a quasi-primary submodule $N$ of $M$ satisfying the primeful property, $\sqrt{(N:M)}$ is a prime ideal of $R$. For the existence of a module-reduced quasi-primary decomposition, the radical of each term appeared in decomposition must be prime. We provide sufficient conditions, involving the saturation and torsion arguments, to ensure that this property holds as is valid in the ideal case. It is proved that for a submodule $N$ of $M$ over a Dedekind domain $R$ which satisffies the primeful property, $N$ is quasi-primary if and only if $radN is prime.

COMMUTATIVE TALL RINGS

A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.

Jónsson and HS Modules over Commutative Rings

LetRbe a commutative ring with identity and letMbe an infinite unitaryR-module. (Unless indicated otherwise, all rings are commutative with identity 1≠0 and all modules are unitary.) ThenMis called aJónsson moduleprovided every proper submodule ofMhas smaller cardinality thanM. Dually,Mis said to behomomorphically smaller(HS for short) if|M/N|<|M|for every nonzero submoduleNofM. In this survey paper, we bring the reader up to speed on current research on these structures by presenting the principal results on Jónsson and HS modules. We conclude the paper with several open problems.

On certain submodules of Weyl modules for SO(2n + 1,𝔽) with char(𝔽) = 2

Abstract. For k = 1 , 2 , ... , n - 1 ${k = 1, 2,\ldots ,n-1}$ let V k = V ( λ k ) ${V_k = V(\lambda _k)}$ be the Weyl module for the special orthogonal group G = SO ( 2 n + 1 , 𝔽 ) ${G = \mathrm {SO}(2n+1,\mathbb {F})}$ with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put V n = V ( 2 λ n ) ${V_n = V(2\lambda _n)}$ . It is well known that all of these modules are irreducible when char(𝔽) ≠ 2 while when char(𝔽) = 2 they admit many proper submodules. In this paper, assuming that char(𝔽) = 2, we prove that Vk admits a chain of submodules V k = M k ⊃ M k - 1 ⊃ ⋯ ⊃ M 1 ⊃ M 0 ⊃ M - 1 = 0 ${V_k = M_k \supset M_{k-1}\supset \dots \supset M_1\supset M_0 \supset M_{-1} = 0}$ where Mi ≅ Vi for 1 , ... , k - 1 ${1,\ldots , k-1}$ and M 0 is the trivial 1-dimensional module. We also show that for i = 1 , 2 , ... , k ${i = 1, 2,\ldots , k}$ the quotient M i / M i-2 is isomorphic to the so-called i-th Grassmann module for G. Resting on this fact we can give a geometric description of M i-1 / M i-2 as a submodule of the i-th Grassmann module. When 𝔽 is perfect, G ≅ Sp(2n,𝔽) and M i / M i-1 is isomorphic to the Weyl module for Sp(2n,𝔽) relative to the i-th fundamental dominant weight of the root system of type Cn . All irreducible sections of the latter modules are known. Thus, when 𝔽 is perfect, all irreducible sections of Vk are known as well.

N-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) n−1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that: (1) If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1. (2) Every n-almost prime submodule of a torsion-free Noetherian module is primary. (3) Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime. (4) There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m 2 = 0. (5) If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

Zariski topology on the spectrum of graded classical prime submodules

Let $R$ be a $G$-graded commutative ring with identity and let $M$ be a graded $R$-module. A proper graded submodule $N$ of $M$ is called graded classical prime if for every $a, b\in h(R)$, $m\in h(M)$, whenever $abm\in N$, then either $am\in N$ or $bm\in N$. The spectrum of graded classical prime submodules of $M$ is denoted by $Cl.Spec_g(M)$. We topologize $Cl.Spec_g(M)$ with the quasi-Zariski topology, which is analogous to that for $Spec_g(R)$.

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

Characterizations of Several Modules Relative to the Class of B(M, X)

Let M and X be right R-modules. We introduce several modules relative to the class of B(M;X) and we investigate relation among these modules. In this note, we show if M is X- -supplemented such that M = M1 M2 implies M1 and M2 are relatively B-projective, then M is an X-H-supplemented module. Throughout this paper, R will be an associative ring with identity, and all mod- ules are unitary right R-modules. A submodule K of M is denoted by K ≤ M. The notation N ≤⊕ M means that N is a direct summand of M. A submodule K of M is called essential (or large) in M (denoted by K ≤e M ), if K ∩ L 0 for every nonzero submodule L of M , and a submodule K of M is called small in M (denoted by K ≪ M ), if N + K M for any proper submodule N of M. A module M is called hollow if every proper submodule of M is small in M. Let N be a submodule of M , a supplement of N in M is a submodule K of M minimal with respect to the property M = N + K, equivalently, M = N + K and N ∩ K ≪ K. Following (14), M is called supplemented if every submodule of M has a supplement in M. M is called a lifting module or (D1)-module if for every submodule A of M there exists a decomposition M = M1 ⊕ M2 such that M1 ≤ A and A ∩ M2 ≪ M2. Following (11), M is called ⊕-supplemented if every submodule of M has a supplement that is a direct summand of M and M is called H-supplemented if for every submodule A of M there is a direct summand D of M such that M = A + X holds if and only if M = D + X. H-supplemented modules are ⊕-supplemented (11, A.2). Suppose N ⊆ K are submodules of M , N is said to be a cosmall submodule of K in M if K/N ≪ M/N (denoted by N ≤ cs K). A submodule N of M is coclosed in M if it has no proper cosmall submodules in M (denoted by N ≤ cc M ). N is called a coclosure of K in M , if N ≤ cs K and N ≤ cc M.

Couniform Modules

In this paper we introduce and study a new concept named couniform modules, which is a dual notion of uniform modules, where an R-module M is said to be couniform if every proper submodule N of M is either zero or there exists a proper submodule N1 of N such that is small submodule of (denoted by ) Also many relationships are given between this class of modules and other related classes of modules. Finally, we consider the hereditary property between R-module M and R-module R in case M is couniform.

Couniform Modules

In this paper we introduce and study a new concept named couniform modules, which is a dual notion of uniform modules, where an R-module M is said to be couniform if every proper submodule N of M is either zero or there exists a proper submodule N1 of N such that is small submodule of (denoted by ) Also many relationships are given between this class of modules and other related classes of modules. Finally, we consider the hereditary property between R-module M and R-module R in case M is couniform.