On Homogeneous Co-maximal Graphs of Groupoid-Graded Rings

Abstract

Let R be a ring with unity which is graded by a cancellative partial groupoid (magma) S. A homogeneous element $0\ne x\in R$ is said to be locally right (left) invertible if there exist an idempotent element $e\in S$ and $x_r\in R$ ($x_l\in R$) such that $xx_r=1_e$ $(x_lx=1_e)$ where $1_e\ne 0$ is a unity of the ring $R_e.$ Element x is said to be locally two-sided invertible if it is both locally right and locally left invertible. The set of all locally invertible elements (left, right, two-sided) of R is denoted by $U_l(R).$ The homogeneous co-maximal graph $\Gamma ^h(R)$of R is defined as a graph whose vertex set consists of all homogeneous elements of R which do not belong to $U_l(R),$ and distinct vertices x and y are adjacent if and only if $xR+yR=R.$ If the edge set of $\Gamma ^h(R)$ is nonempty, then S (with zero) contains a single (nonzero) idempotent element. This condition characterizes the connectedness of $\Gamma ^h(R)\setminus \{0\}$ for a class of groupoid graded rings R which are graded semisimple, graded right Artinian, and which contain more than one maximal graded modular right ideal. If ${\mathbb {F}}_q$ is a finite field and $n\ge 2,$ then the full matrix ring $M_n({\mathbb {F}}_q)$ is naturally graded by a groupoid S with a single nonzero idempotent element. We obtain various parameters of $\Gamma ^h(M_n({\mathbb {F}}_q))\setminus \{0_{M_n({\mathbb {F}}_q)}\}.$ If R is S-graded, with the support equal to $S\setminus \{0\},$ and if $\Gamma ^h(R)\cong \Gamma ^h(M_n({\mathbb {F}}_q)),$ then we prove that R and $M_n({\mathbb {F}}_q)$ are graded isomorphic as S-graded rings.