On 2-primal Ore Extensions

When R is a local ring with a nilpotent maximal ideal, the Ore extension R[x;σ, δ] will or will not be 2-primal depending on the δ-stability of the maximal ideal of R. In the case where R[x;σ, δ] is 2-primal, it will satisfy an even stronger condition; in the case where R[x;σ, δ] is not 2-primal, it will fail to satisfy an even weaker condition. 1. Background and motivation In [13, Proposition 3.7], it was shown that if R is a local, one-sided artinian ring with an automorphism σ, then the skew polynomial ring R[x;σ] must satisfy the (PS I) condition (defined below). The main result of this paper extends [13, Proposition 3.7] to the case of a general Ore polynomial ring R[x;σ, δ]. Recall that in this context, σ is a ring endomorphism ofR, δ is a σ-derivation of R (i.e. a map δ : R −→ R satisfying δ(a+b) = δ(a)+δ(b) and δ(ab) = σ(a)δ(b)+δ(a)b for all a, b ∈ R), and multiplication of polynomials in R[x;σ, δ] (written with lefthand coefficients) is determined by the rule xr = σ(r)x + δ(r) for every r ∈ R. All of our rings will be associative with 1, and generally noncommutative. Let us recall some definitions. A ring R is called 2-primal if the set of nilpotent elements of the ring coincides with the prime radical Nil∗(R). If, more generally, the set of nilpotent elements of a ring is an ideal (not necessarily the prime radical), then the ring is called an NI-ring. A ring R is said to satisfy (PS I) if for every element a ∈ R, the factor ring R/annr (aR) is 2-primal. Taking a = 1 in this definition shows that (PS I) implies 2-primal, whence the following implication chart: (PS I) =⇒ 2-primal =⇒ NI-ring =⇒ general ring In our main result, we will find that the Ore polynomial rings R[x;σ, δ] under consideration will either satisfy the strongest of these four conditions, or else will satisfy only the weakest. This paper appeared in Comm. Algebra 29(5) (2001), 2113–2123. MR 2002e:16042 1