On primitive ideals in polynomial rings over nil rings

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I [x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe’s conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jacobson radical. Our main result states that if R is a nil ring and I an ideal in R[x] (the polynomial ring in one indeterminate over R) then R[x]/I is Jacobson radical if and only if R/I [x] is Jacobson radical, where I ′ is the ideal of R generated by coefficients of polynomials from I. Also if R is a nil ring and I is a primitive ideal of R[x] then I = M [x] for some ideal M of R. It was asked by Beidar, Fong and Puczy lowski [1] whether polynomial rings over nil rings are not (right ) primitive. We show that affirmative answer to this question is equivalent to the Köthe conjecture. We also answer in the negative Question 2 from [1] (Corollary 1). It is known that if a polynomial ring R[x] is primitive then R need not be primitive [3] ( see also Bergman’s example in [5]). Let R be a prime ring and I a nonzero ideal of R. Then R is a primitive ring if and only if I is a primitive ring [6]. Since the Hodges example has a nonzero Jacobson radical it follows that polynomial rings over Jacobson radical rings can be right and left primitive (see also Theorem 3). We recall some definitions after [9] (see also [2], [5]). A right ideal of a ring R is called modular in R if and only if there exists an element b ∈ R such that a− ba ∈ Q for every a ∈ R. If Q is a modular maximal right ideal of R then for every r / ∈ Q, rR +Q = R. An ideal P of a ring R is right primitive in R if and only if there exists a modular maximal right ideal Q of R such that P is the maximal ideal contained in Q. In this paper R[x] denote the polynomial ring in one indeterminate over R. Given polynomial g ∈ R[x] by deg(g) we denote the degree of R, i.e., the