Noetherian and Artinian ordered groupoids—semigroups
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 nec...
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| Format: | Article |
| Language: | English |
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Wiley
2005-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.2041 |
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| author | Niovi Kehayopulu Michael Tsingelis |
| author_facet | Niovi Kehayopulu Michael Tsingelis |
| author_sort | Niovi Kehayopulu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-28881f8e2d6d4d9b89b8c0364998c472 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-28881f8e2d6d4d9b89b8c0364998c4722025-02-03T01:11:23ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-012005132041205110.1155/IJMMS.2005.2041Noetherian and Artinian ordered groupoids—semigroupsNiovi Kehayopulu0Michael Tsingelis1University of Athens, Department of Mathematics, Panepistimiopolis 15784, GreeceUniversity of Athens, Department of Mathematics, Panepistimiopolis 15784, GreeceChain 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.http://dx.doi.org/10.1155/IJMMS.2005.2041 |
| spellingShingle | Niovi Kehayopulu Michael Tsingelis Noetherian and Artinian ordered groupoids—semigroups International Journal of Mathematics and Mathematical Sciences |
| title | Noetherian and Artinian ordered groupoids—semigroups |
| title_full | Noetherian and Artinian ordered groupoids—semigroups |
| title_fullStr | Noetherian and Artinian ordered groupoids—semigroups |
| title_full_unstemmed | Noetherian and Artinian ordered groupoids—semigroups |
| title_short | Noetherian and Artinian ordered groupoids—semigroups |
| title_sort | noetherian and artinian ordered groupoids semigroups |
| url | http://dx.doi.org/10.1155/IJMMS.2005.2041 |
| work_keys_str_mv | AT niovikehayopulu noetherianandartinianorderedgroupoidssemigroups AT michaeltsingelis noetherianandartinianorderedgroupoidssemigroups |