On Homogeneous Co-maximal Graphs of Groupoid-Graded Rings

Let R be a ring with unity which is graded by a cancellative partial groupoid (magma) S. A homogeneous element 0 not equal x is an element of R is said to be locally right (left) invertible if there exist an idempotent element e is an element of S and x(r) is an element of R (X-l is an element of R) such that xx(r) = 1(e) (XlX = 1(e) ) where 1(e) not equal 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) \ {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 F-q is a finite field and n >= 2, then the full matrix ring M-n (F-q) is naturally graded by a groupoid S with a single nonzero idempotent element. We obtain various parameters of Gamma(h)(M-n (F-q)) \ {0M(n) ((Fq))}. If R is S-graded, with the support equal to S\{0}, and if Gamma(h)(R) congruent to Gamma(h)(M-n( )(Fq)), then we prove that R and M-n (F-q) are graded isomorphic as S-graded rings.