PRIMENESS , SEMIPRIMENESS AND PRIME RADICAL OF ORE EXTENSIONS

With the impetus of quantized derivations, renewed interest in the general Ore extension T = R[t;S,D] has arisen during the last few years. The prime radical of T is currently being analyzed (e.g. [G]) and in the special cases when either S = id or D = 0 it has been completely described ([FKM], [PS]). In the former case this description is very much like that for the ordinary polynomial ring, i.e. rad(T ) = I[t; D] where I is a Dideal of R. In the latter case such a description is not possible (cf. [PS]). This difference in behavior is mainly due to the fact that the contraction of an ideal of R[t; D] to R is D-stable but the analogue for R[t; S] and a fortiori for R[t;S, D] is false. One of our aims in Section 5 is to provide some conditions under which rad(T ) is of the form I[t;S,D] where I is an (S, D)-ideal of R. In trying to obtain connections between the prime radical of T and that of the base ring R, different notions appear naturally. These notions (which are of interest on their own) are carefully defined, studied and compared in Sections 1 and 2.