On $(\sigma,\delta) - (S,1)$ rings and their extensions

Vijay Kumar Bhat

Mathematica Moravica, Tome 21 (2017) no. 1.

Résumé

Let R be a ring, $\sigma$ an endomorphism of R and $\delta$ a $\sigma$-derivation of R. We recall that R is called an (S,1)-ring if for $a,b\in R$, $ab = 0$ implies $aRb = 0$. We involve $\sigma$ and $\delta$ to generalize this notion and say that R is a $(\sigma,\delta) - (S,1)$ ring if for $a,b\in R$, $ab = 0$ implies $aRb = 0$, $\sigma(a)Rb = 0$, $aR\sigma(b) = 0$ and $\delta(a)Rb = 0$. In case $\sigma$ is identity, R is called a $δ - (S,1)$ ring. In this paper we study the associated prime ideals of Ore extension $R[x;\sigma,\delta]$ and we prove the following in this direction: Let R be a semiprime right Noetherian ring, which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers), $\sigma$ an automorphism of R and $\delta$ a $\sigma$-derivation of R such that R is a $(\sigma,\delta) - (S,1)$ ring. Then P is an associated prime ideal of $R[x;\sigma,\delta]$ (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R (viewed as a right module over itself) such that $(P \cap R)[x;\sigma,\delta] = P$ and $P \cap R) = U$.

Détail

Mots-clés : Endomorphism, derivation, $(\sigma;\delta) - (S;1)$ ring, Ore extension, prime ideal