Jacobson ring

Abstract: The paper concerns some cases of ring extensions R C S, where S is finitely generated as a right R-module and R is right Noetherian. In ?1 it is shown that if R is a Jacobson ring, then so is S, with the converse true in the PI case. In ?2 we show that if S is semiprime PI, R must also be left (as well as right) Noetherian and S is finitely generated as a left R-module. ?3 contains a result on E-rings. In this paper we collect some results concerning the relationship between rings R c S, where S is finitely generated as a right R-module. For the special cases in which the finite extension S of R is normalizing or centralizing, many theorems have been proved. In this paper we obtain some results of a more general nature. 1. Jacobson property. We call a ring R a Jacobson ring if every prime ideal of R is an intersection of primitive ideals. Let J(R) denote the Jacobson radical of R and N(R) the lower nilradical, that is, the intersection of all the prime ideals of R. With this notation, the alternative formulations of the Jacobson property are: J(R/P) = 0 for all prime ideals P of R. J(R) = N(R) for every homomorphic image R of R. If R is PI or right Noetherian, then every nil ideal is contained in N(R), and the last condition is equivalent to: J(R) is nil for every homomorphic image R of R. THEOREM 1. Let R be a right Noetherian subring of a ring S, such that S is finitely generated as a right R-module. Then, if R is Jacobson, so must S be Jacobson. PROOF. Given any prime ideal P of S, we want to show that J(S/P) = 0. Since the hypothesis of the theorem holds for the ring embedding R/P n R C S/P, the problem reduces to the case where S is a prime ring. We therefore want to show that, if S is prime, then J(S) = 0. Suppose J(S) $ 0. By a standard result on Goldie rings, J(S) must contain a regular element, say a. Since SR is finitely generated and R is right Noetherian, there is a positive integer n, such that the elements 1, a, a2,... ,an are integrally dependent over R. That is, for some rn-,. . . , ri, rO E R, we have an +an-lrn-1 + * + ar1 + ro = 0. If n is minimal, then, since a is regular, ro $ 0. But then rO E J(S) n R. Hence J(S) n R $ 0. We claim that J(S) nR c J(R). Let x E J(S) nR and suppose x V J(R). Then there is an element r E R for which 1 rx is not invertible in R. But 1 rx is Received by the editors March 31, 1987. 1980 Mathematics Subject (Classification (1985 Revision). Primary 16A38, 16A33, 16A21. ?D1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Abstract: In this article we consider rings R with a partial action α of an infinite cyclic group G on R. We generalize the well-known results about Jacobson rings and strongly Jacobson rings in skew polynomial rings and skew Laurent polynomial rings to partial skew polynomial rings and partial skew Laurent polynomial rings.

Abstract: We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can characterize the maximum number of analytically independent elements in an arbitrary ideal of a local ring and that dim B is not greater than dim A if B is a subalgebra of A and A is a (not necessarily finitely generated) subalgebra of a finitely generated algebra over a Noetherian Jacobson ring.