A matrix ring 
is the ring of all 
matrices with entries in
a ring 
, using matrix addition
and matrix multiplication. If 
has a multiplicative
identity, the identity matrix is the identity
of 
. For 
and nonzero 
with identity, 
is noncommutative even when 
is commutative (Lam 2001).
When 
has a multiplicative
identity, the ring center of 
consists of the scalar matrices 
with 
in the ring center of 
. The commuting
graph of a noncommutative matrix ring records the pairs of noncentral matrices
that commute with each other.
