A Note on Comaximal Graph of Non-commutative Rings

Abstract

"If R is a commutative ring with unity and the graph Γ2(R)\J(R) is n-partite, then the number of maximal ideals of R is at most n." The proof of this result is not correct. In this paper we present a correct proof for this result. Also we generalize some results given in the aforementioned paper for the non-commutative rings.

On graph associated to co-ideals of commutative semirings

“…In [11], Sharma and Bhatwadekar defined another graph on a ring R with vertices as elements of R and there is an edge between two distinct vertices x and y in R if and only if Rx + Ry = R. Further, in [10], Maimani et al studied the graph defined by Sharma and Bhatwadekar and called it comaximal graph. Also, in [1], Akbari et al studied the comaximal graph over non-commutative ring.…”

Co-maximal ideal graphs of matrix algebras

“…There are a lot of papers which apply combinatorial methods to obtain algebraic results in ring theory (see [1,4,12]). …”

“…In [12], authors defined the co-maximal graph associated with a commutative ring R, (R), whose vertex set consists all elements of R and two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Also the co-maximal graph of a non-commutative ring was studied in [1,14]. An ideal version of co-maximal graphs of commutative rings can be found in [2,13].…”

Comaximal graph of $C(X)$

“…In [27], comaximal graph of a commutative ring was defined. On later, in [21], [26], [16], [28], [24], [22], [19], [2], [3], [1], [30], and [23], this investigation was continued.…”