On the essential annihilating-ideal graph of commutative rings

On the cozero-divisor graphs associated to rings

“…Various algebraic graphs associated to ring structures have been defined and investigated by the researchers (see, [1,4,5,8,9,15,19,20]). Because of the vital role of ideals in the theory of rings, numerous authors have studied the graphs on rings with respect to their ideals such as inclusion ideal graph [2], intersection graphs of ideals [10], ideal-relation graph [14], prime ideal sum graph [23], reduced cozero-divisor graph [35], co-maximal ideal graph [36] etc.…”

Coloring in essential annihilating-ideal graphs of commutative rings

“…(2) If R is non-reduced, then the following statements are equivalent: (2) (i) ⇒ (ii) Suppose that EG(R) = K n , for some positive integer n. By [4, Theorem 1.4], R is Artinian. Now, the result follows from [13,Theorem 3.6].…”

“…If Nil(R) is a minimal ideal, then it is adjacent to every other vertex and since ω(EG(R)) = 2, EG(R) is star. By [13,Theorem 3.5], EG(R) = K 1,∞ , in this case. If Nil(R) is not minimal, then it contains exactly one ideal Nil(R) 2 .…”

“…The essential annihilating-ideal graph of a ring R is defined as the graph EG(R) and two distinct vertices I, J are joined by an edge if and only if Ann(IJ) ≤ e R. This graph was first introduced and studied by Nazim and Rehman in [13] as the ideal version of essential graph (see [15]). The authors proved many interesting results for EG(R).…”

“…In[13, Theorem 3.7], it has been claimed that "for a ring R with at least one minimal ideal we haveEG(R) = K m,n ,where m, n ≥ 2 if and only if R = D × S, where D, S are two integral domains which are not fields". It should be noted that if R = D × S, where D, S are two integral domains which are not fields, then R does not have any minimal ideal, in other words, Soc(R) = 0.…”