  An interchange ring,(R,+,*)is an abelian group with a second binary operation
defined so that the interchange law (x+y)*(u+v)=(x*u)+(y*v)holds. An
interchange near ring is the same structure based on a group which may not be
abelian. It is shown that given any group,G,each interchange(near)ring based on
that group is formed from a pair of endomorphisms of G whose images commute,
and that all interchange (near)rings based on G can be characterized in this
manner. To obtain an associative interchange ring, the endomorphisms must be
commuting idempotents in the endomorphism semigroup of G. When G is abelian we
develop a group theoretic analogue of the simultaneous diagonalization of
idempotent linear operators and show that pairs of endomorphisms which yield
associative interchange rings can be diagonalized and then put into a canonical
form. A best possible upper bound of 4^r can be given for the number of
distinct isomorphism classes of associative interchange rings based on a finite
abelian group A which is a direct sum of r cyclic groups of prime power order.
If A is direct sum of r copies of the same cyclic group of prime power order,
we show that there are exactly (r+1)(r+2)(r+3)/6 distinct isomorphism classes
of associative interchange rings based on A. Several examples are given and
further comments are made about the general theory of interchange rings.
