International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 13, Pages 2041-2051
doi:10.1155/IJMMS.2005.2041
Noetherian and Artinian ordered groupoids—semigroups
University of Athens, Department of Mathematics, Panepistimiopolis 15784, Greece
Received 9 February 2005; Revised 16 June 2005
Copyright © 2005 Niovi Kehayopulu and Michael Tsingelis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Chain conditions, finiteness conditions, growth conditions, and
other forms of finiteness, Noetherian rings and Artinian rings
have been systematically studied for commutative rings
and algebras since 1959. In pursuit of the deeper results of ideal
theory in ordered groupoids (semigroups), it is necessary to study
special classes of ordered groupoids (semigroups). Noetherian
ordered groupoids (semigroups) which are about to be
introduced are particularly versatile. These satisfy a
certain finiteness condition, namely, that every ideal of the
ordered groupoid (semigroup) is finitely generated. Our purpose
is to introduce the concepts of Noetherian and Artinian ordered
groupoids. An ordered groupoid is said to be Noetherian if every
ideal of it is finitely generated. In this paper, we prove that an
equivalent formulation of the Noetherian requirement is that the
ideals of the ordered groupoid satisfy the so-called ascending
chain condition. From this idea, we are led in a natural way to
consider a number of results relevant to ordered groupoids with
descending chain condition for ideals. We moreover prove that an
ordered groupoid is Noetherian if and only if it satisfies the
maximum condition for ideals and it is Artinian if and only if it
satisfies the minimum condition for ideals. In addition, we prove
that there is a homomorphism π of an ordered groupoid
(semigroup) S having an ideal I onto the Rees
quotient ordered groupoid (semigroup) S/I. As a
consequence, if S is an ordered groupoid and I an ideal of
S such that both I and the quotient groupoid S/I are Noetherian (Artinian), then so is S. Finally, we give
conditions under which the proper prime ideals of commutative
Artinian ordered semigroups are maximal ideals.