Restricted Minimum Condition in Reduced Commutative Rings

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10452372" target="_blank" >RIV/00216208:11320/22:10452372 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=L1sGzd7Qnx" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=L1sGzd7Qnx</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00009-022-02190-4" target="_blank" >10.1007/s00009-022-02190-4</a>

Result language
angličtina
Original language name
Restricted Minimum Condition in Reduced Commutative Rings
Original language description
We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, the factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains the RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. In the second part, we will study CDR domains, i.e., domains where for any two ideals I, J the inclusion I subset of J implies that I is a multiple of J. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials R[x] is RM if and only if R is a reduced Artinian ring. And we will study the relation between RM rings and UFDs.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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