Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Bhat, Vijay Kumar. "Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings." Czechoslovak Mathematical Journal 63.4 (2013): 1049-1056. <http://eudml.org/doc/260773>.

@article{Bhat2013,
abstract = {Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma $ be an automorphism of $R$ and $\delta $ a $\sigma $-derivation of $R$. Then $R$ is said to be an almost $\delta $-divided ring if every minimal prime ideal of $R$ is $\delta $-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb \{Q\}$ ($\mathbb \{Q\}$ is the field of rational numbers). Let $\sigma $ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta $ a $\sigma $-derivation of $R$ such that $\sigma (\delta (a)) = \delta (\sigma (a))$ for all $a \in R$. Further, if for any strongly prime ideal $U$ of $R$ with $\sigma (U) = U$ and $\delta (U)\subseteq \delta $, $U[x; \sigma , \delta ]$ is a strongly prime ideal of $R[x; \sigma , \delta ]$, then we prove the following: (1) $R$ is a near pseudo valuation ring if and only if the Ore extension $R[x; \sigma ,\delta ]$ is a near pseudo valuation ring. (2) $R$ is an almost $\delta $-divided ring if and only if $R[x;\sigma ,\delta ]$ is an almost $\delta $-divided ring.},
author = {Bhat, Vijay Kumar},
journal = {Czechoslovak Mathematical Journal},
keywords = {Ore extension; automorphism; derivation; minimal prime; pseudo-valuation ring; near pseudo-valuation ring; Ore extensions; automorphisms; derivations; minimal prime ideals; strongly prime ideals; near pseudo-valuation rings},
language = {eng},
number = {4},
pages = {1049-1056},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings},
url = {http://eudml.org/doc/260773},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Bhat, Vijay Kumar
TI - Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 1049
EP - 1056
AB - Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma $ be an automorphism of $R$ and $\delta $ a $\sigma $-derivation of $R$. Then $R$ is said to be an almost $\delta $-divided ring if every minimal prime ideal of $R$ is $\delta $-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb {Q}$ ($\mathbb {Q}$ is the field of rational numbers). Let $\sigma $ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta $ a $\sigma $-derivation of $R$ such that $\sigma (\delta (a)) = \delta (\sigma (a))$ for all $a \in R$. Further, if for any strongly prime ideal $U$ of $R$ with $\sigma (U) = U$ and $\delta (U)\subseteq \delta $, $U[x; \sigma , \delta ]$ is a strongly prime ideal of $R[x; \sigma , \delta ]$, then we prove the following: (1) $R$ is a near pseudo valuation ring if and only if the Ore extension $R[x; \sigma ,\delta ]$ is a near pseudo valuation ring. (2) $R$ is an almost $\delta $-divided ring if and only if $R[x;\sigma ,\delta ]$ is an almost $\delta $-divided ring.
LA - eng
KW - Ore extension; automorphism; derivation; minimal prime; pseudo-valuation ring; near pseudo-valuation ring; Ore extensions; automorphisms; derivations; minimal prime ideals; strongly prime ideals; near pseudo-valuation rings
UR - http://eudml.org/doc/260773
ER -