Uppers to zero in $R[x]$ and almost principal ideals

Title:

Uppers to zero in $R[x]$ and almost principal ideals

Creator:

Borna, Keivan and Mohajer-Naser, Abolfazl

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:804c4a06-25a5-46f1-9abd-a54900a7a6a9
uuid:804c4a06-25a5-46f1-9abd-a54900a7a6a9

Subject:

almost principal ideal, divisorial ideal, greatest common divisor domain, Schreier domain, and uppers to zero

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let $R$ be an integral domain with quotient field $K$ and $f(x)$ a polynomial of positive degree in $K[x]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I = f(x)K[x] \cap R[x]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1} = R$. – $I^{-1}$ as the $R[x]$-submodule of $K(x)$ is of finite type. Furthermore we prove that for $I = f(x)K[x] \cap R[x]$ we have: – $I^{-1}\cap K[x]=(I:_{K(x)}I)$. – If there exists $p/q \in I^{-1}-K[x]$, then $(q,f)\neq 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I' = q(x)K[x] \cap R[x]$ is an almost principal upper to zero. Finally we show that a Schreier domain $R$ is a greatest common divisor domain if and only if every upper to zero in $R[x]$ contains a primitive polynomial.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

565-572

Source:

Czechoslovak Mathematical Journal | 2013 Volume:63 | Number:2

Harvested from:

CDK

Metadata only:

false