For a RING R, the intersection minimal IDEAL graph, denoted by ∧(R), is a simple undirected graph whose vertices are proper non-zero (right) IDEALs of R and any two distinct vertices I1 and I2 are adjacent if and only if I1 ∩ I2 is a minimal IDEAL of R. In this article, we explore connectedness, clique number, split character, planarity, independence number, domination number of ∧(R).

Let R be a commutative RING with non-zero identity. The annihilatorinclusion IDEAL graph of R, denoted by  R, is a graph whose vertex set is the of all non-zero proper IDEALs of R and two distinct vertices I and J are adjacent if and only if either Ann(I)  J or Ann(J)  I. The purpose of this paper is to provide some basic properties of the graph  R. In particular, shows that  R is a connected graph with diameter at most three, and has girth 3 or 1. Furthermore, is determined all isomorphic classes of non-local Artinian RINGs whose annihilator-inclusion IDEAL graphs have genus zero or one.

Let R be a commutative RING with identity. We use j (R) to denote the comaximal IDEAL graph. The vertices of j (R) are proper IDEALs of R which are not contained in the Jacobson radical of R, and two vertices I and J are adjacent if and only if I+J=R. In this paper we show some properties of this graph together with planarity of line graph associated to j (R).

Let R be a commutative RING with unity. The comaximal IDEAL graph of R, denoted by C (R), is a graph whose vertices are the proper IDEALs of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. In this paper, we classify all comaximal IDEAL graphs with finite independence number and present a formula to calculate this number. Also, the domination number of C (R) for a RING R is determined. In the last section, we introduce all planar and toroidal comaximal IDEAL graphs. Moreover, the commutative RINGs with isomorphic comaximal IDEAL graphs are characterized. In particular we show that every finite comaximal IDEAL graph is isomorphic to some C (Zn).

The concepts of fuzzy semi-IDEALs of R with respect to H£R and generalized fuzzy quotient RINGs are introduced. Some properties of fuzzy semi IDEALs are discussed. Finally, several isomorphism theorems for generalized fuzzy quotient RINGs are established.

The annihilating-IDEAL graph of a commutative RING R is denoted by AG (R), whose vertices are all nonzero IDEALs of R with nonzero annihilators and two distinct vertices I and Jare adjacent if and only if IJ=0. In this article, we completely characterize RINGs R when gr (AG (R)) ¹3.

‎The graph $ AG ( R ) $ {of} a commutative RING $R$ with identity has an edge linking two unique vertices when the product of the vertices equals {the} zero IDEAL and its vertices are the nonzero annihilating IDEALs of $R$‎.‎The annihilating-IDEAL graph with {respect to} an IDEAL $ ( I ) $, ‎which is {denoted} by $ AG_I ( R ) $‎, ‎has distinct vertices $ K $ and $ J $ that are adjacent if and only if $ KJ\subseteq I $‎. ‎Its vertices are $ \{K\mid KJ\subseteq I\ \text{for some IDEAL}\ J \ \text{and}\ K$‎, ‎$J \nsubseteq I‎, ‎K\ \text{is a IDEAL of}\ R\} $‎. ‎The study of the two graphs $ AG_I ( R ) $ and $ AG(R/I) $ and {extending certain} prior findings are two main objectives of this research‎. ‎This studys {among other things‎, ‎the} findings {of this study reveal}‎‎that $ AG_I ( R ) $ is bipartite if and only if $ AG_I ( R ) $ is triangle-free‎.

ConsideRING the fact that the words in the intra-religious attitude are considered in two ways, the first is towards jurisprudential words and the second is a rational and conceptual view focusing on their meanings. we have tried to deal with one of the most important issues of intra-religious attitude with a rational approach by descriptive and analytical method, which is the category of "hermeneutic interpretation" of words, especially the most influential idea of interpretation & that is the view of the "spirit of meaning" of Ghazali. After Ghazali, Ibn Arabi used this theory of Ghazali regarding the development of meaning in creating his new idea, and by re-reading Ghazali's idea with a mystical approach, he presented his own theory regarding the expansion of meaning. While accepting the argument of the focal point of the theory of the spirit of meaning, he removes it from the exclusivity to the "linear", and for this purpose, he brought up arguments as well as the Conditions of Hermeneutic Interpretation in this regard. Ibn Arabi's arguments on transversal interpretation include new formulations in the argument, which after him, this theory was favored and accepted by Mulla Sadra, and then it was accepted by his students.

The RINGs considered in this article are commutative with identity which admit at least two maximal IDEALs. This article is inspired by the work done on the comaximal IDEAL graph of a commutative RING. Let R be a RING. We associate an undirected graph to R denoted by G(R), whose vertex set is the set of all proper IDEALs I of R such that I ̸  J(R), where J(R) is the Jacobson radical of R and distinct vertices I1; I2 are adjacent in G(R) if and only if I1 \ I2 = I1I2. The aim of this article is to study the interplay between the graph-theoretic properties of G(R) and the RING-theoretic properties of R.