(Jordan) derivation on amalgamated duplication of a ring along an ideal

Let A be a ring and I be an ideal of A. The amalgamated duplication of A along I is the subring of A × A defined by $A\bowtie I := {(a, a + i) |a ∈ A, i ∈ I}$. In this paper, we characterize $A\bowtie I$ over which any (resp. minimal) prime ideal is invariant under any derivation provided that A is semiprime. When A is noncommutative prime, then $A\bowtie I$ is noncommutative semiprime (but not prime except if I = (0)). In this case, we prove that any map of $A\bowtie I$ which is both Jordan and Jordan triple derivation is a derivation.

Generalizations of S- Prime Ideals

Let R be a commutative ring with identity and S be a multiplicative subset of R . In this paper we introduce the concept of almost S-prime ideal as a new generalization of S−prime ideal. Let P be a proper ideal of R disjoint with S. Then P is said to be almost S- prime ideal if there exists s ∈ S such that, for all x, y ∈ R if xy ∈ P − P 2 then sx ∈ P or sy ∈ P. Number of results concerning this concept and examples are given. Furthermore, we investigate an almost S- prime ideals of trivial ring extensions and amalgamation rings..

Graded 1-absorbing prime ideals of graded commutative rings

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

A note on almost prime submodule of CSM module over principal ideal domain

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

Fuzzy Congruence Relations in Generalized Almost Distributive Fuzzy Lattices

This paper defines the fuzzy congruence relation of GADFL (Generalized nearly distributive fuzzy lattices). The ideas of θ - ideal and θ - Prime ideal are introduced in GADFL, and the fuzzy congruence relation is used to explain these ideals. AMS subject classification: 06D72, 06F15, 08A72.

N Sequence Prime Ideals

In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

On Graded Weakly S-Prime Ideals

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.

Intuitionistic Fuzzy Normed Prime Ideal and Some of Their Characteristics

This Paper intends to introduce the notion of prime ideals of Intuitionistic fuzzy normed Rings and to establish basic properties related to it. It investigates these notions and shown anew Result using intuinistic fuzzy points and non membership function incorparating with t-norm and s-norm to show aome results of fuzzy prime ideal.