In this study, all rings are commutative with non-zero identity and all modules are considered to be unital. Let $M$ be a left $R$-module. A proper submodule $N$ of $M$ is called an $S$-$weakly$ $prime$ submodule if $0_{M}\neq f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in S=End(M)$ and $m\in M$. Some results concerning $S$-prime and $S$-weakly prime submodules are obtained. Then we study $S$-prime and $S$-weakly prime submodules of multiplication modules. Also for $R$-modules $M_{1}$ and $M_{2},$ we examine $S$-prime and $S$-weakly prime submodules of $M=M_{1}\times M_{2},$ where $S=S_{1}\times S_{2},$ $S_{1}=End(M_{1})$ and $S_{2}=End(M_{2})$.
In this study, we aim to introduce the concept of a 1-absorbing prime submodule of an unital module over a commutative ring with a non-zero identity. Let M be an R-module and N be a proper submodule of M. For all non-unit elements a, b in R and m in M if abm in N, either ab in (N : M) or m in N, then N is called 1-absorbing prime submodule of M. We show that the new concept is a generalization of prime submodules at the same time it is a kind of special 2-absorbing submodule. In addition to some properties of a 1-absorbing prime submodule, we obtain a characterization of it in a multiplication…
Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $ϕ: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $ϕ-$prime $R$-submodule of $M$ if whenever $r$ is a homogeneous element of $R$ and $m$ is a homogeneous element of $M$ such that $rm\in K-ϕ(K)$, then either $m\in K$ or $r\in (K:_{R}M)$. If $ϕ(K)=\emptyset$ for all $K\in GS(M)$, then a graded $ϕ-$prime submodule is exactly a graded prime submodule. If $ϕ(K)=…
Let $G$ be a group with identity $e$, $R$ be a commutative $G$-graded ring with unity $1$ and $M$ be a $G$-graded unital $R$-module. In this article, we introduce the concept of graded $1$-absorbing prime submodule. A proper graded $R$-submodule $N$ of $M$ is said to be a graded $1$-absorbing prime $R$-submodule of $M$ if for all non-unit homogeneous elements $x, y$ of $R$ and homogeneous element $m$ of $M$ with $xym\in N$, either $xy\in (N :_{R} M)$ or $m\in N$. We show that the new concept is a generalization of graded prime submodules at the same time it is a special graded $2$-absorbing su…
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 classical prime submodule, if for each $m \in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$ or $bm\in N$. We introduce the concept of "weakly classical prime submodules." A proper submodule $N$ of $M$ is a weakly classical prime submodule if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then $am\in N$ or $bm\in N$.
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. By $J(R),$ we denote the Jacobson radical of $R$. The purpose of this paper is to introduce the concept of weakly $J$-submodules generalizing $J$-submodules. We call a proper submodule $N$ of $M$ a weakly $J$-submodule if whenever $0\neq rm\in N$ for $r\in R$ and $m\in M$, then $r\in(J(R)M:M)$ or $m\in N.$ Various properties and characterizations of weakly $J$-submodules are investigated especially in the case of multiplication modules.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $I_{e}$-prime submodule as a generalization of a graded prime submodule for $I=\oplus_{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning of these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded $I_{e}$-prime submodule of $M$ if whenever $% r_{g}\in h(R)$ and $m_{h}\in h(M)$ with $r_{g}m_{h}\in N-I_{e}N,$ then either $r_…
Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=ϕ$ to be weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. Many properties, examples and characterizations of weakly $S$-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotie…
The purpose of the present paper is to prove some properties of the strongly irreducible submodules in the arithmetical and Noetherian modules over a commutative ring. The relationship among the families of strongly irreducible submodules, irreducible submodules, prime submodules and primal submodules is proved. Also, several new characterizations of the arithmetical modules are given. In the case when $R$ is Noetherian and $M$ is finitely generated, several characterizations of strongly irreducible submodules are included. Among other things, it is shown that when $N$ is a submodule of $M$ su…
Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-primary if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in\sqrt{(N:_{R}M)}$ or $sm\in N$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, q…
In this paper, we introduce the concepts of strongly 2-absorbing primary ideals (resp., submodules) and strongly 2-absorbing ideals (resp., submodules) as generalizations of strongly prime ideals. Furthermore, we investigate some basic properties of these classes of ideals.
In this article, we introduce the concepts of graded $s$-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $s$-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of grading by $\mathbb{Z}$, a finite group, or a polycyclic-by-finite group, in addition…
We introduce the notion of Burch submodules and weakly $\mathfrak m$-full submodules of modules over local rings and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test property. We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or $M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-full in $X$, is 1-Tor rigid and a test module provided that $X$ is faithful (and $X/M$ has finite length when $M$ is weakly $\mathfrak m$-full). As an application, we …
Let $R$ be a commutative ring with identity, $S$ a multiplicatively closed subset of $R$, and $M$ be an $R$-module. In this paper, we study and investigate some properties of $S$-primary submodules of $M$. Among the other results, it is shown that this class of modules contains the family of primary (resp. $S$-prime) submodules properly.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.
Let R be a commutative ring with identity. A prime submodule P of an R-module M is called coprimely structured if, whenever P is coprime to each element of an arbitrary family of submodules of M, the intersection of the family is not contained in P. An R-module M is called coprimely structured provided each prime submodule of M is coprimely structured. In this paper, properties of coprimely structured modules are examined. Severals results for coprimely structured finitely generated modules and coprimely structured multiplication modules are obtained.
In this paper, we give a generalization for weakly primary submodules called $I$-primary submodule and we study some properties of it. We give some characterizations of $I$-primary submodules. Also we establish the situation of $I$-primary submodules in module localizations and decomposition modules.
Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We obtain useful characterizations for those modules M for…
Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We prove that if AG(M) is a tree, then either AG(M) is a s…
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring, $M$ a graded $R$-module and $A\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $A$-2-absorbing submodules of $M$ as a generalization of graded 2-absorbing submodules and graded $A$-prime submodules of $M.$ We investigate some properties of this class of graded submodules.
This paper is concerned with S-co-m modules which are a generalization of co-m modules. In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is a multiplicatively closed subset of $R$. We prove that if $M$ is an S-co-m module satisfying the S-DAC and $N\leq M$, then $N\leq ^{S}_{e}M$ if and only if there exists $I\ll ^{S}R$ such that $s(0:_{M}I)\leq N\leq (0 :_{M}I)$ for some $s\in S$. Let $M$ be a faithful S-strong co-m $R$-module. We prove that if $N\ll ^{S}M$ then there exists an ideal $I\leq ^{S…
This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each nonunits $x, y, z \in A$ with $0\neq xyz \in P$, then either $xy \in P$ or $z \in P$. In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in $C(X)$, which is the ring of continuous f…
Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spec_G^s(M)$ and it is called the graded second spectrum of $M$. In this paper, we discuss graded rings with Noetherian graded prime spectrum and obtain some conclusions. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. Using these conclusions and properties, we also investigate $Spec_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space and give some related outco…
It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each …
Let $R$ be a $G$-graded ring. In this article, we introduce two new concepts on graded rings, namely, weakly graded rings and invertible graded rings, and we discuss the relations between these concepts and several properties of graded rings. Also, we study the concept of weakly crossed products and study some properties defined on weakly crossed product and give the relationship between this new concept and several properties of graded rings. Moreover, in this article, we give a generalization for the concept of graded essential submodules, and introduce the concept of graded semi-uniform mod…
Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. Let $Γ_{\mathfrak{a}}$ denote the $\mathfrak{a}$-torsion functor. Conditions are given for the (weakly) associated primes of $Γ_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/Γ_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.
In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in P-IP$ implies either $a \in P$ or $b \in P$. We give some characterizations of $I$-prime ideals and study some of its properties. Moreover, we give conditions under which $I$-prime ideals becomes prime or weakly prime and we construct the view of $I$-prime ideal in decomposite rings.
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_…
