(untitled) (European Journal of Pure and Applied Mathematics, 2015)
In this paper, we discuss a stronger type of primary decomposition (known as transparency) in noncommutative set up. One of the class of noncommutative rings are the skew polynomial rings. We show that certain skew polynomial rings satisfy this type of primary decomposition.Â Recall that a right Noetherian ring R is said to be transparent ring if there exist irreducible ideals I_j, 1 ≤ j ≤ n such that _j = 1^nI_j = 0 and each R/I_j has a right artinian quotient ring.Â Let R be a commutative Noetherian ring, which is also an algebra over Q (Q is the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Then we show that the skew polynomial ring R[x;σ,δ] is a transparent ring.