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

Title:

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

Creator:

Bhat, Vijay Kumar

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:b1233a1f-3e8e-4708-9efd-fd94c867cf09
uuid:b1233a1f-3e8e-4708-9efd-fd94c867cf09

Subject:

Ore extension, automorphism, derivation, minimal prime, pseudo-valuation ring, and near pseudo-valuation ring

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

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

Coverage:

1049-1056

Source:

Czechoslovak Mathematical Journal | 2013 Volume:63 | Number:4

Harvested from:

CDK

Metadata only:

false