Let R be a commutative ring.
All its endomorphisms form a monoid ℰ(R) and a natural question to ask is what
monoids appear as full endomorphism monoids of commutative rings. It was shown in
[8] that every group is representable as the full automorphism group of a ring
without unit element. Much more cannot be expected in this case as the
zero mapping is always one of the endomorphisms. The presence of the unit
element 1 in the ring changes the picture. We will show here that every monoid
is isomorphic to the monoid ℰ1(R) of all 1-preserving endomorphisms of
a commutative ring R with 1. In fact, a stronger theorem will be proved:
the category ℛ1 of all rings with 1 and all 1-preserving homomorphisms is
binding.
