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In this paper, the notion of fully idempotent modules is defined and it is shown that this notion inherits most of the essential properties of the usual notion of von Neumann's regular rings. Furthermore, we introduce the dual notion of fully idempotent modules (that is, fully coidempotent modules) and investigate some properties of this class of modules.Keywords: Idempotent submodule, fully idempotent module, coidempotent submodule, copure submodule, fully coidempotent module
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Journal of Algebraic Structures and Their Applications, Volume:2 Issue: 1, Winter - Spring 2015, PP 35 -47In this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. Several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given.
Then we investigate for a ring R, when any finite intersection of (uniformly) primary submodules of any R-module is a (uniformly) classical quasi-primary submodule. Furthermore, the behavior of classical quasi-primary and uniformly classical quasi-primary submodules under localizations are studied. Also, we investigate the existence of (minimal) primary submodules containing classical quasi-primary submodules.Keywords: Classical quasi-primary, Uniformly classical quasi-primary -
International Journal Of Nonlinear Analysis And Applications, Volume:12 Issue: 1, Winter-Spring 2021, PP 2231 -2241
In this paper, the concept of fuzzy visible submodules which is a new type of fuzzy submodules has been introduced. Some results and characterizations of fuzzy visible are established namely the homomorphic image of the fuzzy visible submodule, the sum of two fuzzy visible submodules. The relation between fuzzy visible submodule and its submodules. Also, the fuzzy quotient modules in sense of fuzzy visible have been presented. We prove that the intersection of a collection of fuzzy visible submodules are visible submodules and the converse is not true. Also, we define the strong cancellation fuzzy modules and we established some results of it with respect to fuzzy visible submodules. Many other properties we study in fuzzy visible submodules.
Keywords: Fuzzy visible submodule, Pure fuzzy submodule, T-pure fuzzy submodule, Fully cancellation fuzzy module, Strongly cancellation fuzzy module -
The concept of essential submodules is a well known concept.In this paper we try to replace an arbitrary submodule of M,say T, instead of 0 in the definition of essential submodules. By this,essential submodules are precisely {0}-essential submodules. For a submoduleK of right R-module M, we have K ess M if and only if(K : m) is annr(m)-essential right ideal of R, for each m 2 M \ {0}.Among other things, this generalization of essential submodules gives anecessary and sufficient condition for MT being finitely co-generated.
Keywords: Essential submodules, s-essential submodules, socle of a module -
In this paper, our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$. Let $T$ be an arbitrary submodule of an $R$-module $M$, then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$, whenever for every submodule $X$ of $M$, $N\cap X\subseteq T$ implies that $(T:M)\subseteq ^{e}{\rm Ann}(X)$. We investigate some new results concerning to this class of submodules. Among various results we prove that for a faithful multiplication $R$-module $M$, if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$, then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$. The converse is true if $M$ is moreover a finitely generated module.Keywords: multiplication module, faithful module, essential submodule
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Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly. In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful property.Keywords: Primary, Primary, like submodule, like submodule, weakly primary, like submodule, primeful property, weakly primary, multiplication module
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Journal of Algebraic Structures and Their Applications, Volume:8 Issue: 1, Winter Spring 2021, PP 61 -73
In this article we study and investigate the behavior of $r$-submodules (a proper submodule $N$ of an $R$-module $M$ in which $amin N$ with ${rm Ann}_M(a)=(0)$ implies that $min N$ for each $ain R$ and $min M$). We show that every simple submodule, direct summand, divisible submodule, torsion submodule and the socle of a module is an $r$-submodule and if $R$ is a domain, then the singular submodule is an $r$-submodule. We also introduce the concepts of $uz$-module (i.e., an $R$-module $M$ such that either ${rm Ann}_M(a)not=(0)$ or $aM=M$, for every $ain R$) and strongly $uz$-module (i.e., an $R$-module $M$ such that $aMsubseteq a^2M$, for every $ain R$) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly $uz$-module and every Artinian $R$-module is a $uz$-module. It is observed that if $M$ is a faithful cyclic $R$-module, then $M$ is a $uz$-module if and only if every its cyclic submodule is an $r$-submodule. In addition, in this case, $R$ is a domain if and only if the only $r$-submodule of $M$ is zero submodule. Finally, we prove that $R$ is a $uz$-ring if and only if every faithful cyclic $R$-module is a $uz$-module.
Keywords: $r$-submodule, $uz$-module, strongly $uz$-module, $r$-ideal -
In this paper, we introduce the concept of 2-absorbing semiprimary submodules in modules over a commutative ring with nonzero identity which is a generalization of 2-absorbing primary submodule. Let N be a proper submodule of an R-module M. Then N is said to be a 2-absorbing semiprimary submodule of M if whenever a1a2 ∈ R, m ∈ M and a1a2m ∈ N, then a1a2 ∈ (N :R M) or a1m ∈ N or an 2 m ∈ N, for some positive integer n. We have given an example and proved number of results concerning 2-absorbing semiprimary submodules.
Keywords: 2-absorbing semiprimary submodule, 2-absorbing primary submodule, 2-absorbing primary ideal, primary idea -
In this paper, we introduce the concepts of a fuzzy essential-small submodule and a fuzzy small-essential submodule of a module. We investigate various properties of such fuzzy submodules. It is also shown that the Jacobson L-radical is the sum of all essential-small L-submodules of a module. We also prove that the L-socle is the intersection of all small-essential L-submodules of a module.
Keywords: Fuzzy small submodule, fuzzy essential submodule, fuzzy small-essential andfuzzy essential-small submodule -
Let R be a commutative ring with identity. A proper submodule N of an R-module M is an n-submodule if rm ∈ N (r ∈ R, m ∈ M) with r /∈ p AnnR(M), then m ∈ N. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.
Keywords: n-ideal, n-submodule
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