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); an automorphism of R and a -derivation of R such that ((a)) = ((a)), for all a ∈ R. Further more if a(a) ∈ P(R) implies that a ∈ P(R), (P(R) the prime radical of R), then R[x;,] is a Transparent ring.