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

Borna, Keivan, and Mohajer-Naser, Abolfazl. "Uppers to zero in $R[x]$ and almost principal ideals." Czechoslovak Mathematical Journal 63.2 (2013): 565-572. <http://eudml.org/doc/260617>.

@article{Borna2013,
abstract = {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 $(q,f)\ne 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I^\{\prime \} = q(x)K[x] \cap R[x]$ 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.},
author = {Borna, Keivan, Mohajer-Naser, Abolfazl},
journal = {Czechoslovak Mathematical Journal},
keywords = {almost principal ideal; divisorial ideal; greatest common divisor domain; Schreier domain; uppers to zero; greatest common divisor domain; uppers to zero},
language = {eng},
number = {2},
pages = {565-572},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Uppers to zero in $R[x]$ and almost principal ideals},
url = {http://eudml.org/doc/260617},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Borna, Keivan
AU - Mohajer-Naser, Abolfazl
TI - Uppers to zero in $R[x]$ and almost principal ideals
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 565
EP - 572
AB - 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 $(q,f)\ne 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I^{\prime } = q(x)K[x] \cap R[x]$ 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.
LA - eng
KW - almost principal ideal; divisorial ideal; greatest common divisor domain; Schreier domain; uppers to zero; greatest common divisor domain; uppers to zero
UR - http://eudml.org/doc/260617
ER -