Restricted Minimum Condition in Reduced Commutative Rings

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Restricted Minimum Condition in Reduced Commutative Rings

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Restricted Minimum Condition in Reduced Commutative Rings

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

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

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Mediterranean Journal of Mathematics

ISSN

1660-5446

e-ISSN

1660-5454

Svazek periodika

19

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

10

Strana od-do

253

Kód UT WoS článku

000868460900002

EID výsledku v databázi Scopus

2-s2.0-85139858920