Uppers to zero in $R[x]$ and almost principal ideals
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 565-572
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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.
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.
DOI :
10.1007/s10587-013-0038-9
Classification :
13A05, 13A15, 13B25, 13F15
Keywords: almost principal ideal; divisorial ideal; greatest common divisor domain; Schreier domain; uppers to zero
Keywords: almost principal ideal; divisorial ideal; greatest common divisor domain; Schreier domain; uppers to zero
@article{10_1007_s10587_013_0038_9,
author = {Borna, Keivan and Mohajer-Naser, Abolfazl},
title = {Uppers to zero in $R[x]$ and almost principal ideals},
journal = {Czechoslovak Mathematical Journal},
pages = {565--572},
year = {2013},
volume = {63},
number = {2},
doi = {10.1007/s10587-013-0038-9},
mrnumber = {3073979},
zbl = {06236432},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0038-9/}
}
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 SP - 565 EP - 572 VL - 63 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0038-9/ DO - 10.1007/s10587-013-0038-9 LA - en ID - 10_1007_s10587_013_0038_9 ER -
%0 Journal Article %A Borna, Keivan %A Mohajer-Naser, Abolfazl %T Uppers to zero in $R[x]$ and almost principal ideals %J Czechoslovak Mathematical Journal %D 2013 %P 565-572 %V 63 %N 2 %U http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0038-9/ %R 10.1007/s10587-013-0038-9 %G en %F 10_1007_s10587_013_0038_9
Borna, Keivan; Mohajer-Naser, Abolfazl. Uppers to zero in $R[x]$ and almost principal ideals. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 565-572. doi: 10.1007/s10587-013-0038-9
Cité par Sources :