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- Title
Restricted iso-minimum condition.
- Authors
Daneshvar, Asghar
- Abstract
A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l , where each D i is a principal ideal domain and each A i is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 픭 , and R / 픭 is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided.
- Subjects
ARTIN rings; NOETHERIAN rings; COMMUTATIVE rings; PRIME ideals; LOCAL rings (Algebra); HOMOMORPHISMS; ISOMORPHISM (Mathematics)
- Publication
Forum Mathematicum, 2023, Vol 35, Issue 5, p1211
- ISSN
0933-7741
- Publication type
Article
- DOI
10.1515/forum-2022-0225