On maximal ideals and the brown-mccoy radical of polynomial rings

Abstract

In this paper we obtain some results on maximal ideals of polynomial rings R[x] in one indeterminate x. In particular we complete Ferrero's characterization [5] of rings R with an identity such that R[x] contains a maximal R-disjoint ideal, i.e., a maximal ideal M satisfying M n R = 0. We also get several results on the Brown-McCoy radical of R[x]. Recall that for a given ring A the Brown-McCoy radical U(A) of A is defined as the intersection of all ideals I of A such that A/I is a simple ring with an identity. In particular a ring is Brown-McCoy radical if and only if it cannot be homomorphically mapped onto a ring with an identity, or equivalently, onto a simple ring with an identity. In [7] Krernpa proved that for every ring R, U(R[xJ) = (U(R[x]) n R)[xJ. We shall show that U(R[x]) n R is equal to the intersection of all prime ideals I of R such that the centre of R/I has a non-zero intersection with each non-zero ideal of RII. In particular, if R is a nil ring, then R[x] is Brown-McCoy radical, i.e., R[x] cannot be homomorphically mapped onto