  In this paper we introduce compressed commuting graph of rings. It can be
seen as a compression of the standard commuting graph (with the central
elements added) where we identify the vertices that generate the same subring.
The compression is chosen in such a way that it induces a functor from the
category of rings to the category of graphs, which means that our graph takes
into account not only the commutativity relation in the ring, but also the
commutativity relation in all of its homomorphic images. Furthermore, we show
that this compression is best possible for matrix algebras over finite fields.
We compute the compressed commuting graphs of finite fields and rings of $2
\times 2$ matrices over finite fields.
