Transparent Ore Extensions over $sigma$(∗)-rings

Abstract

In this paper we introduce a stronger type of primary decomposition of a Noetherian ring. We call such a ring a Transparent ring and show that if R is a commutative Noetherian ring, which is also an algebra over Q (the field of rational numbers); $sigma$ an automorphism of R and $delta a sigma$-derivation of R such that$delta (sigma(a))=sigma(delta(a)), $ for all a ∈ R. Further more if $a sigma (a)$ ∈ P(R) implies that a ∈ P(R), (P(R) the prime radical of R), then R$[x;delta, sigma]$ is a Transparent ring.