Vol. 45, No. 2, 1973

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Homomorphisms of commutative rings with unit element

Ervin Fried and J. Sichler

Vol. 45 (1973), No. 2, 485–491

Let R be a commutative ring. All its endomorphisms form a monoid (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 1(R) of all 1-preserving endomorphisms of a commutative ring R with 1. In fact, a stronger theorem will be proved: the category 1 of all rings with 1 and all 1-preserving homomorphisms is binding.

Mathematical Subject Classification 2000
Primary: 13A99
Secondary: 20M15
Received: 15 June 1971
Published: 1 April 1973
Ervin Fried
J. Sichler