In this study, we introduce a new concept of n-hyperideals of a commutative multiplicative hyperring. Let (R, +, o) be a commutative multiplicative hyperring. Here, the multiplication is a hyperoperation, while the addition is a binary operation. Let o be a hyperoperation from R x R -> P & lowast;(R), where P & lowast; (R) denotes the set of all non-empty subset of R. A proper hyperideal I of R is said to be an n-hyperideal if a o b subset of I and a is not an element of N (R), then b is an element of I for all a, b is an element of R. In addition to giving some main properties, we obtain a characterization of an n-hyperideal of R. Moreover, we give some examples and investigate its special properties.

We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let R be a commutative ring with a nonzero identity and Q a proper ideal of R. The proper ideal Q is said to be a weakly strongly quasi-primary ideal if whenever 0 ≠ ab ∈ Q for some a, b ∈ R, then a2 ∈ Q or $$b \in \sqrt Q $$ . Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero dimensional rings over which every proper ideal is wsq-primary. Finally, we study finite union of wsq-primary ideals.

Abstract In this paper, we study S-Principal ideal multiplication modules. Let be a commutative ring with a multiplicatively closed set and an A-module. A submodule N of M is said to be an S-multiple of M if there exist and a principal ideal I of A such that . is said to be an S-principal ideal multiplication module if every submodule of is an S-multiple of M. Various examples and properties of S-principal ideal multiplication modules are given. We investigate the conditions under which the trivial extension is an -principal ideal ring. Also, we prove Cohen type theorem for S-principal ideal multiplication modules in terms of S-prime submodules.

Throughout this paper, $R$ is an associative ring (not necessarily commutative) with identity and $M$ is a right $R$-module with unitary. In this paper, we introduce a new concept of $\phi$-prime submodule over an associative ring with identity. Thus we define the concept as following: Assume that $S(M)$ is the set of all submodules of $M$ and $\phi:S(M)\rightarrow S(M)\cup\{\emptyset\}$ is a function. For every $Y\in S(M)$ and ideal $I$ of $R,$ a proper submodule $X$ of $M$ is called $\phi$-prime, if $YI\subseteq X$ and $YI\nsubseteq\phi(X),$ then $Y\subseteq X$ or $I\subseteq(X:_{R}M)$. Then we examine the properties of $\phi$-prime submodules and characterize it when $M$ is a multiplication module.

Recall that a commutative ring [Formula: see text] is a locally integral domain if its localization [Formula: see text] is an integral domain for each prime ideal [Formula: see text] of [Formula: see text] Our aim in this paper is to extend the notion of locally integral domains to modules. Let [Formula: see text] be a commutative ring with a unity and [Formula: see text] a nonzero unital [Formula: see text]-module. [Formula: see text] is called a locally torsion-free module if the localization [Formula: see text] of [Formula: see text] is a torsion-free [Formula: see text]-module for each prime ideal [Formula: see text] of [Formula: see text] In addition to giving many properties of locally torsion-free modules, we use them to characterize Baer modules, torsion free modules, and von Neumann regular rings.

Let R be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of n-ideals and the class of (2, n)-ideals. A proper ideal I of R is said to be a quasi n-ideal if √(I) is an n-ideal of R. Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the n-ideals, the quasi primary ideals, the (2, n)-ideals and the pr-ideals. Moreover, we use the quasi n-ideals to characterize some kind of rings. Finally, we investigate quasi n-ideals under various contexts of constructions such as direct product, power series, idealization, and amalgamation of a ring along an ideal.

In this paper, we introduce the concept of 2-absorbing submodule elements in an le-module M as follows: a proper submodule element q in M is said to be 2-absorbing for any r,s is an element of R and m is an element of M if rsm <= q, then either rs is an element of (q : e) or rm <= q or sm <= q. Moreover, we define some generalizations of the new concept such as weakly 2-absorbing, n-absorbing, weakly n-absorbing, (n,k)-absorbing, weakly (n,k)-absorbing submodule elements in le-modules. After presenting a main example for le-modules, we study some counter examples for the generalizations. In addition to giving some characterizations for the new concepts, we investigate the relationship between prime (primary) submodule elements and them.

In this study, we present the generalization of the concept of $r$-ideals in commutative rings with nonzero identity. Let $R$ be a commutative ring with $0\neq1$ and $L(R)$ be the lattice of all ideals of $R$. Suppose that $\phi:L(R)\rightarrow L(R)\cup\left\{\emptyset\right\}$ is a function. A proper ideal $I$ of $R$ is called a $\phi-r$-ideal of $R$ if whenever $ab\in I$ and $Ann(a)=(0)$ imply that $b\in I$ for each $a,b\in R.$ In addition to giving many properties of $\phi-r$-ideal, we also examine the concept of $\phi-r$-ideal in trivial ring extension and use them to characterize total quotient rings.

This paper aims to introduce a new class of submodules, called (m, n)-semiprime submodule, which is a generalization of semiprime submodule. Let M be a unital A-module and m,n is an element of N. Then a proper submodule P of M is said to be an (m, n)-semiprime submodule if whenever a(m)x is an element of P for some a is an element of A,x is an element of M, then a(n)x is an element of P. In addition to giving many characterizations and properties of this kind of submodules, we also use them to characterize von Neumann regular modules.

In this article, we introduce and study the concept of $\phi$-2-absorbing quasi primary ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $L(R)$ be the lattice of all ideals of $R$. Suppose that $\phi:L(R)\rightarrow L(R)\cup\left\{ \emptyset\right\} $ is a function. A proper ideal $I$ of $R$ is called a $\phi$-2-absorbing quasiprimary ideal of $R$ if $a,b,c\in R$ and whenever $abc\in I-\phi(I),$ then either $ab\in\sqrt{I}$ or $ac\in\sqrt{I}$ or $bc\in\sqrt{I}$. In addition to giving many properties of $\phi$-2-absorbing quasi primary ideals, we also use them to characterize von Neumann regular rings.

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 module.

This paper deals with the pure elements and the dual notions of prime elements (that is, second elements). For this, it introduces the definitions of second element and coprime element. Then it is shown that the concepts of the second element and coprime element are equivalent. Moreover, this study gives us a characterization of comultiplication modules. Finally, it defines pure elements and obtains the relation among pure, idempotent and multiplication elements.

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 work, we propose a novel key reconciliation protocol for the quantum key distribution (QKD). Based on Newton's polynomial interpolation, the proposed protocol aims to correct all erroneous bits at the receiver without revealing information to the eavesdropper. We provide the exact frame error rate (FER) expression of the proposed protocol. The inherent nature of the proposed algorithm ensures correcting all erroneous bits if the algorithm succeeds. We present an information-theoretical proof that the revealed information during the key reconciliation process is equal to zero. We also provide a numerical comparison of our algorithm with the asymptotic performance of the error-correcting codes and two exemplary low-density-parity-check (LDPC) codes. The results highlight that our algorithm provides superior performance when compared to the LDPC codes, regardless of the distance between Alice and Bob. Furthermore, the proposed key reconciliation protocol is usable for the longer quantum link distances than the state-of-the-art protocols.

In this study, we introduce phi-2-absorbing and phi-2-absorbing primary submodules of modules over commutative rings generalizing the concepts of 2-absorbing and 2-absorbing primary submodules. Let phi : S(M) -> S(M) boolean OR {phi} be a function where S(M) denotes the set of all submodules of M and N a proper submodule of an R-module M. We will say that N is a phi-2-absorbing submodule of M if whenever a, b is an element of R, m is an element of M with abm is an element of N and abm (sic) phi(N), then am is an element of N or bm is an element of N or ab is an element of (N :(R) M) and N is said to be a phi-2-absorbing primary submodule of M whenever if a, b is an element of R, m is an element of M with abm is an element of N and abm (sic) phi(N), then am is an element of M-rad(N) or bm is an element of M-rad(N) or ab is an element of (N :(R) M). We investigate many properties of these new types of submodules and establish some characterizations for phi-2-absorbing and phi-2-absorbing primary submodules of multiplication modules.

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 is an element of M and elements a, b is an element of R, abm is an element of N implies that am is an element of N or bm is an element of N. Let phi : S(M) -> S(M) U {empty set} be a function where S(M) is the set of all submodules of M. We introduce the concept of "phi-classical prime submodules". A proper submodule N of M is a phi-classical prime submodule if whenever a, b is an element of R and m is an element of M with abm is an element of N\phi(N), then am is an element of N or bm is an element of N.

In this paper we investigate delta- primary submodules which unify prime submodules and primary submodules. Our motivation is to extend the concept of delta- primary ideals into delta- primary submodules of modules over commutative rings. A number of main results about prime and primary submodules are extended into this general framework.

In this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module. With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.

In this paper, we introduce the concept of φ-2-absorbing elements in multiplicative lattices. Letφ : L → L ∪ {∅} be a function. We will say a proper element q of L to be a φ-2-absorbing element of L if whenevera, b, c ∈ L with abc ≤ q andabc φ(q) implies either ab ≤ q or ac ≤ q or bc ≤ q. We give some basic properties and establish some characterizations of φ-2-absorbing elements in some special lattices.

Let R be a commutative ring with 1 not equal 0 and S(R) be the set of all ideals of R. In this paper, we extend the concept of 2-absorbing primary ideals to the context of 0-2-absorbing primary ideals. Let phi : S(R) -> S(R) U null set be a function. A proper ideal I of R is said to be a phi-2-absorbing primary ideal of R if whenever a, b, c is an element of R with abc is an element of I - phi (I) implies ab is an element of I or ac is an element of root I or be E. A number of results concerning phi-2-absorbing primary ideals are given.