It has been known that the graph isomorphism problem is
polynomial-time many-one reducible to the ring isomorphism problem. 
In fact, two different reductions have already been proposed.
For those reductions, 
rings of certain types have been used to represent a given graph.
In this paper, we give yet another reduction,
which is based on a simpler and more natural construction of a ring from a graph.
By the existing reductions, one of the original graph isomorphisms can be found
in each ring isomorphism obtained for the reduced ring isomorphism problem instance.
On the other hand, in our new reduction,
it is not clear how to get a graph isomorphism between two graphs
from an obtained ring isomorphism between rings constructed from the graphs.
However, we show that we can compute a graph isomorphism
from an obtained ring isomorphism in polynomial time.
In fact, one ring isomorphism may correspond to many graph isomorphisms in our reduction.
Our proof essentially shows a way to obtain all graph isomorphisms
corresponding to one ring isomorphism.