Narrow Results

Let R be a commutative ring and M be an R-module with a proper submodule N. The total graph of M with respect to N, denoted by T(Γ_N(M)), is investigated. The vertex set of this graph is M and for all x, y belonging to M, x is adjacent to y if and only if x + y ∈ M(N), where M(N) = {m ∈ M : rm ∈ N for some r ∈ R - (N : M)}. In this paper, in addition to studying some algebraic properties of M(N), we investigate some graph theoretic properties of two important subgraphs of T(Γ_N(M)) in the cases depending on whether or not M(N) is a submodule of M.

A notion of 2-primal rings is generalized to modules by defining 2-primal modules. We show that the implications between rings which are reduced, have insertion-of-factor-property (IFP), reversible, semi-symmetric and 2-primal are preserved when the notions are extended to modules. Like for rings, 2-primal modules bridge the gap between modules over commutative rings and modules over non-commutative rings; for instance, for 2-primal modules, prime submodules coincide with completely prime submodules. Completely prime submodules and reduced modules are both characterized. A generalization of 2-primal modules is done where 2-primal and NI modules are a special case.

It is well known that the concept of left serial ring is a Morita invariant property and a theorem due to Nakayama and Skornyakov states that “for a ring $R$, all left $R$-modules are serial if and only if $R$ is an Artinian serial ring”. Most recently the notions of “prime uniserial modules” and “prime serial modules” have been introduced and studied by Behboodi and Fazelpour in [Prime uniserial modules and rings, submitted; Noetherian rings whose modules are prime serial, Algebras and Represent. Theory19(4) (2016) 11 pp]. An $R$-module $M$ is called prime uniserial ($\wp$-uniserial) if its prime submodules are linearly ordered with respect to inclusion, and an $R$-module $M$ is called prime serial ($\wp$-serial) if $M$ is a direct sum of $\wp$-uniserial modules. In this paper, it is shown that the $\wp$-serial property is a Morita invariant property. Also, we study what happens if, in the above Nakayama–Skornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every $\wp$-serial module is serial. Let $R$ be Morita equivalent to a commutative ring $S$. It is shown that every $\wp$-uniserial left $R$-module is uniserial if and only if $R$ is a zero-dimensional arithmetic ring with $J(R)$ T-nilpotent. Moreover, if $S$ is Noetherian, then every $\wp$-serial left $R$-module is serial if and only if $R$ is serial ring with dim$(R)\le 1$.

Let $R$ be an arbitrary ring and $M$ be a nonzero right $R$-module. In this paper, we introduce the set $T_R(M)=\{m\in M|mI=0$, for some nonzero ideal $I$ of $R\}$ of strong torsion elements of $M$ and the properties of this set are investigated. In particular, we are interested when $T_R(M)$ is a submodule of $M$ and when it is a union of prime submodules of $M$.

Let $R$ be a commutative ring with identity, $Q$ be a prime ideal of $R$ and $F$ be a free $R$-module of finite rank. In this paper, we use hypergraphs to give a full characterization of prime submodules of $F$. Also we define a hypergraph $P{H}_Q(F)$ called the prime submodules hypergraph of $F$ with respect to $Q$ and use properties of Steiner systems for counting the number of $Q$-prime submodules of $F$ when the number of cosets of $Q$ in $R$ is finite.

Let M be an R-module. If for every submodule N of M, there exists an element r ∈ R such that N=rM, then we say that M is a principal ideal multiplication module. In this paper, the relations between principal ideal multiplication modules, multiplication modules, cyclic modules, and modules over principal ideal rings are studied. It is proved that every principal ideal multiplication module over any quotient of a Dedekind domain is cyclic. Also, every principal ideal multiplication module with prime annihilator ideal is cyclic.

Let R be an associative ring with identity and M an R-module. Let Spec(M) be the set of all prime submodules of M. We topologize Spec(M) with the Zariski topology and prove some useful results.

In this paper, we extend the concept of Ako and Oka families to submodules, study the behavior of the extended prime submodule principle and use these concepts to give new proofs of some familiar theorems.

We introduce the notion of prime submodules of a given right R-module and describe all properties of them as a generalization of prime ideals in associative rings.

A right R-module M is called a Goldie module if it has finite Goldie dimension and satisfies the ACC for M-annihilator submodules of M. In this paper, we study the class of prime Goldie modules and the class of semiprime Goldie modules as generalizations of prime right Goldie rings and semiprime right Goldie rings.

In this paper, we introduce the notion of almost prime submodules of a given right $R$-module and study some properties of them as a generalization of weakly prime ideals in associative rings. We will also investigate modules in which every fully invariant submodule is almost prime.