Polynomial rings over nil rings need not be nil (Q1593789)

The author constructs a non-nil polynomial ring over a nil ring, solving a long standing problem of S. A. Amitsur. She takes the algebra \(A\) of polynomials with \(0\) constant terms in noncommuting indeterminates \(x,y,z\) over a countable field \(K\), enumerates the elements of \(A\), and considers the ideal \(I\) of \(A\) generated by rapidly increasing powers of these elements. Then the factor algebra \(R=A/I\) is nil. She proves then that for every countable field \(K\) there is a nil algebra \(R\) over \(K\) such that the polynomial algebra \(R[x,y]\) in commuting indeterminates \(x,y\), is not nil. Corollaries: There exists a nil algebra \(N\) such that the polynomial algebra \(N[x]\) is not nil; there exists a nil algebra \(N\) such that the polynomial algebra \(N[x,y]\) in two commuting indeterminates is not Jacobson radical.