Radicals Of Polynomial Rings

Abstract

Introduction. Let R be a ring and let R [x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R [x]) and L(R [x]). The Jacobson radical J(R [x]) is shown to be a polynomial ring N [x] over a nil ideal N of R and the lower radical L(R [x]) is the polynomial ring L(R) [x].