On Primitive Ideals in Polynomial Rings over Nil Rings

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] 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 afterA 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 1

Some results in noncommutative ring theory

“…[15]; however, this ring is not nil. Note that the polynomial ring in one indeterminate over Sasiada's ring is left and right primitive ( [44]). By Nakayama's lemma a simple Jacobson radical ring cannot be finitely generated.…”

A Note on Primitivity of Ideals in Skew Polynomial Rings of Automorphism Type

“…A ring is said to be a Jacobson ring if every prime ideal of is an intersection of (either left or right) primitive ideals of . In [1] Now we recall some terminology and results; see [2][3][4]. A right ideal of a ring is called modular in if and only if there exists an element ∈ such that − ∈ for every ∈ .…”

“…Following [1] we have the following. We say that V is a "good number for ," if, for all sufficiently large , there are ∈ [ ; ] such that − ∈ with deg( ) ≤ V. Let ⊆ ; we denotẽ …”

On the Behrens radical of matrix rings and polynomial rings

“…In [2] this approximation was improved to become the Behrens radical, which lies between the Jacobson and Brown-McCoy radicals. Recently this was improved further in [17] by showing that if I is a primitive ideal in the polynomial ring R[x] in one indeterminate x over a nil ring R (the existence of such an ideal is equivalent to a negative solution of Köthe's problem), then I has the form of "polynomials over an ideal of R", namely, I = J [x] for some ideal J of R.…”