Semigroup Forum Vol. 62 (2001) 311–316 c  2001 Springer-Verlag New York Inc. DOI: 10.1007/s002330010057 RESEARCH ARTICLE Commutative Ideal Extensions of Abelian Groups J. C. Rosales, P. A. Garc´ ıa-S´ anchez, and J. I. Garc´ ıa-Garc´ ıa Communicated by John Fountain Introduction Ideal extensions of semigroups were introduced by Clifford in [1] and since then they have been widely studied (see for instance [2]). Our aim here is to characterize commutative ideal extensions of Abelian groups. We show that they are those commutative semigroups with an idempotent Archimedean element, or equivalently, those commutative semigroups E such that E/R is an Abelian group, where R is the least congruence that makes E/R cancellative. These characterizations give rise to an algorithm for deciding from a presentation of a finitely generated commutative monoid whether it is an ideal extension of an Abelian group. Finally we present a procedure that enables us to compute the set of idempotents of a finitely generated commutative monoid. All semigroups appearing in this paper are commutative and for this reason in the sequel we sometimes omit the adjective commutative whenever we refer to a commutative semigroup (the same holds for monoids). The authors would like to thank the referee for his/her comments and suggestions. 1. Commutative ideal extensions of Abelian groups Let S be a semigroup. An ideal extension of S is a semigroup E fulfilling that S is one of its ideals. In this section we characterize semigroups that are ideal extensions of Abelian groups. An element x in a semigroup S is an idempotent if 2x = x . The element x ∈ S is Archimedean if for every y ∈ S there exist k ∈ N \{0} and z ∈ S such that kx = y + z (where N denotes the set of nonnegative integers). Theorem 1.1. Let E be a commutative semigroup. Then E is an ideal ex- tension of an Abelian group if and only if E has an element that is Archimedean and idempotent.