On Primitive Ideals in Polynomial Rings over Nil Rings
Agata Smoktunowicz2005, Algebras and Representation Theory
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Abstract
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
