L-Fuzzy Semiprime Ideals of a Poset

In this paper, we introduce the concept of L-fuzzy semiprime ideal in a general poset. Characterizations of L-fuzzy semiprime ideals in posets as well as characterizations of an L-fuzzy semiprime ideal to be L-fuzzy prime ideal are obtained. Also, L-fuzzy prime ideals in a poset are characterized.

The image of Jordan left derivations on algebras

Let A be an algebra, and let I be a semiprime ideal of A. Suppose thatd : A → A is a Jordan left derivation such that d(I) ⊆ I.We prove that if dim{d(a)+I : a ⋲ A} ≤ 1, then d(A) ⊆ I. Additionally, we consider several consequences of this result.

On Hoehnke Ideal in Ordered Semigroups

For a proper subset $A$ of an ordered semigroup $S$, we denote by $H_A(S)$ the subset of $S$ defined by $H_A(S):=\{h\in S \mbox { such that if } s\in S\backslash A, \mbox { then } s\notin (shS]\}$. We prove, among others, that if $A$ is a right ideal of $S$ and the set $H_A(S)$ is nonempty, then $H_A(S)$ is an ideal of $S$; in particular it is a semiprime ideal of $S$. Moreover, if $A$ is an ideal of $S$, then $A\subseteq H_A(S)$. Finally, we prove that if $A$ and $I$ are right ideals of $S$, then $I\subseteq H_A(S)$ if and only if $s\notin (sI]$ for every $s\in S\backslash A$. We give some examples that illustrate our results. Our results generalize the Theorem 2.4 in Semigroup Forum 96 (2018), 523--535.

Semiprime Ideals of \(\Gamma\)-So-Rings

A \(\Gamma\)-so-ring is a structure possessing a natural partial ordering, an infinitary partial addition and a ternary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional composition is a \(\Gamma\)-so-ring. In this paper we introduce the notions of semiprime ideal and \(p\)-system in \(\Gamma\)-so-rings and we obtain the characteristics between them.

COMPLEMENT OF THE ZERO DIVISOR GRAPH OF A LATTICE

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.

A note on Cohn's universal localisation at a semiprime ideal

The universal localisation RΓ(s) at a semiprime ideal S of a left Noetherian ring R was defined and studied by P. M. Cohn. In this note we investigate the interaction between the universal localisation RΓ(s), the Ore localisation at S, and the torsion-theoretic localisation at the injective envelope E(R/S) of the module R(R/S).

A Characterization of semiprime ideals in near-rings

It is well known that in any near-ring, any intersection of prime ideals is a semiprime ideal. The aim of this paper is to prove that any semiprime ideal I in a near-ring N is the intersection of all minimal prime ideals of I in N. As a consequence of this we have any seimprime ideal I is the intersectionof all prime ideals containing I.

Factoring Ideals into Semiprime Ideals

Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.