POLYNOMIAL-RINGS OVER GOLDIE-KERR COMMUTATIVE RINGS

All rings in this paper are commutative, and acc perpendicular-to (resp., acc +) denotes the acc on annihilators (resp., On direct sums of ideals). Any subring of an acc perpendicular-to ring, e.g., of a Noetherian ring, is an acc perpendicular-to ring. Together, acc perpendicular-to and acc + constitute the requirement for a ring to be a Goldie ring. Moreover, a ring R is Goldie iff its classical quotient ring Q is Goldie. A ring R is a Kerr ring (the appellation is for J. Kerr, who in 1990 constructed the first Goldie rings not Kerr) iff the polynomial ring R[x] has acc perpendicular-to (in which case R must have acc perpendicular-to) . By the Hilbert Basis theorem, if S is a Noetherian ring, then so is S[x]; hence, any subring R of a Noetherian ring is Kerr. In this note, using results of Levitzki, Herstein, Small, and the author, we show that any Goldie ring R such that Q = Q(c)(R) has nil Jacobson radical (equivalently, the nil radical of R is an intersection of associated prime ideals) is Kerr in a very strong sense: Q is Artinian and, hence, Noetherian (Theorems 1.1 and 2.2). As a corollary we prove that any Goldie ring A that is algebraic over a field k is Artinian, and, hence, any order R in A is a Kerr ring (Theorem 2.5 and Corollary 2.6). The same is true of any algebra A over a field k of cardinality exceeding the dimension of A (Corollary 2.7). Other Kerr rings are: reduced acc perpendicular-to rings and valuation rings with acc perpendicular-to (see 3.3 and 3.4).