ideals

In this paper, we introduce a modified version of the simple-intersection graph for semisimple rings, applied to a ring $R$ with unity. The findings from this modified version are subsequently utilized to solve several coloring optimization problems.  We demonstrate how the clique number of the simple-intersection graph can be used to determine the maximum number  of possibilities that can be selected from a set of $n$ colors without replacement or order, subject to the constraint that  any pair shares only one common color. We also show how the domination number can be used to determine the  minimum number of possibilities that can be selected, such that any other possibility shares one color with  at least one of the selected possibilities, is $n-1$.

The notions of dense, proper, separated or perfect morphisms and hence of compact, Hausdorff or compact Hausdorff are all consequent to good properties of a family of closed morphisms is well known in literature. Deeper consequences like the Tychonoff product theorem or the Stone Čech compactifications follow from richer properties of the set of closed morphisms. The purpose of this paper is to provide a closure operation on a preneighbourhood space so that the resulting set of closed morphisms possess all the properties mentioned above.

In this research work, we introduce a generalization of the notion of kernel of a set in topological spaces endowed with an ideal, which is a fundamental tool to obtain new modifications of open sets and closed sets. Using this generalized kernel, we define and characterize new low separation axioms in other contexts obtained from a topological space endowed with an ideal. Also, we study the invariance of these low separation axioms under certain types of continuity defined in this novel theoretical framework.

In this paper, we introduce the notion of hybrid bi-ideals in semigroups and investigate some of their important properties. We also give various equivalent conditions for a semigroup to be regular and hybrid structures to be hybrid bi-ideals of S.

In this paper, we define the conceps of complex fuzzy Lie subalgebras and complex fuzzy ideals of Lie algebras with respect to t-norms and investigate some of characteristics and relationship between them. Next, we introduce the concepos of quotient subalgebras, intersection, sum and direct product of them and prove some results about them. Finally, we introduce and study the image and the inverse image of them under Lie algebra homomorphisms.

The objectives of this article are to present the notions of anti complex fuzzy subalgebras and anti complex fuzzy ideals of Lie algebras under  $S$-norms and we prove that the level subset of them are also subalgebras and ideals of Lie algebras, respectively. Next, we introduce the union and summation of them and we prove that the union and summation of them are also anti complex fuzzy subalgebras and anti complex fuzzy ideals of Lie algebras under  $S$-norms, respectively.  Finally, we investigate the homomorphic image (pre-image)  of them under Lie homomorphisms.

Let $\mathcal{A},$ $\mathcal{B},$ and $\mathcal{X}$ be complex algebras, $\theta : \mathcal{B} \longrightarrow \mathcal{X}$ be an algebra homomorphism, and let $\mathcal{A}$ be an $\mathcal{X}$-bimodule. We define a product on $\mathcal{A} \times \mathcal{B}$ as $(a_1, b_1)(a_2,b_2) = (a_1 a_2 + a_1 \cdot \theta(b_2) + \theta(b_1) \circ a_2,b_1 b_2)$ for all $(a_1, b_1), (a_2,b_2) \in \mathcal{A} \times \mathcal{B}$ and write $\mathcal{A} \times \mathcal{B}$ with this product by $\mathcal{A} \times_\theta\mathcal{B}$. We shall study some basic properties of $\mathcal{A} \times_\theta \mathcal{B}$. When $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{X}$ are Banach algebras, $\mathcal{A}$ is a Banach $\mathcal{X}$-bimodule, and $\theta$ is a continuous homomorphism with the norm at most $1$, we determine the ideals of $\mathcal{A} \times_\theta \mathcal{B}$ of a certain type, the Gelfand space of this Banach algebra, and the module multipliers of this Banach algebra.

The annihilating-ideal graph of a commutative ring $R$ with unity is defined as the graph $AG(R)$ whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ = 0$. Nikandish et.al. proved that $AG(mathbb{Z}_n)$ is weakly perfect. In this short paper, we characterize $n$ for which $AG(mathbb{Z}_n)$ is perfect.

This paper deals with some results concerning the notion of extended ideal based zero divisor graph Γ ¯ ¯ ¯ I (R) for an ideal I of a commutative ring R and characterize its bipartite graph. Also, we study the properties of an annihilator of Γ¯ Γ¯I(R)

In this note, we study the lattice structure on the class of all weak hyper K-ideals of a hyper K-algebra. We first introduce the notion of (left,right) scalar in a hyper K-algebra which help us to characterize the weak hyper K-ideals generated by a subset. In the sequel, using the notion of a closure operator, we study the lattice of all weak hyper K ideals of a hyper K-algebra, and we prove a special subclass of this class together with the suitable operations forms a Boolean lattice.

In this paper, we introduce the notion of hyper pseudo B C K - algebras, which is a generalization of pseudo BCK -algebras and hyper BCK -algebras and we investigates some related properties. In follow, we de ne some kinds of hyper pseudo BCK -ideals of a hyper pseudo BCK - algebra and we find the relations among them. Finally, we characterize the hyper pseudo BCK -ideals of type 4 generated by a nonempty subset.

In this paper, we introduce the notion of primal strong co-ideals and give some results involving them. It is shown that subtractive strong co-ideals are intersection of all primal strong co-ideals that contain them. Also we prove that the representation of strong co-ideals as reduced intersections of primal strong co-ideals is unique.

This paper consider the general forms of $(\alpha,\beta)$-fuzzy left ideals (right ideals, bi-ideals, interior ideals) of an ordered semigroup, where $\alpha,\beta\in\{\in_{\gamma},q_{\delta},\in_{\gamma}\wedgeq_{\delta}, \in_{\gamma}\vee q_{\delta}\}$ and $\alpha\neq \in_{\gamma}\wedge q_{\delta}$. Special attention is paid to $(\in_{\gamma},\ivq)$-left ideals (right ideals, bi-ideals, interior ideals) and some related properties are investigated. The characterization of regular ordered semigroups in terms of $(\in_{\gamma},\ivq)$-fuzzy left (right) ideals, $(\in_{\gamma},\ivq)$-fuzzy bi-ideals and $(\in_{\gamma},\ivq)$-fuzzy interior ideals is also investigated.