Generating Adjoint Groups

We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

Nilpotent, Algebraic and Quasi-Regular Elements in Rings and Algebras

We prove that an integral Jacobson radical ring is always nil, which extends a well-known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial px with integer coefficients, such that px(1) = 1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the Köthe conjecture: namely, the integral rings.

Derivations and bounded nilpotence index

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS

Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc.7 (1956) 35–48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x-1] are nil-Armendariz relative to any u.p.-monoid M, and we have nil (R[M]) = nil (R)[M].

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

On Primitive Ideals in Graded Rings

Let R = be a graded nil ring. It is shown that primitive ideals in R are homogeneous. Let A = be a graded non-PI just-infinite dimensional algebra and let I be a prime ideal in A. It is shown that either I = ﹛0﹜ or I = A. Moreover, A is either primitive or Jacobson radical.

RADICALS WITH THE α-AMITSUR PROPERTY

A radical γ of rings is said to have the Amitsur property if for all rings A, γ(A[X]) = (γ(A[X]) ∩ A)[X]. Let Xα denote a set of indeterminates of cardinality α. We say that γ has the α-Amitsur property if for all rings A, γ(A[Xα]) = (γ(A[Xα]) ∩ A)[Xα]. We study properties of this type of radicals and show relationships with other known radicals for rings. A ring A is said to be an absolute γ-ring if A[x1,…, xn] ∈ γ, for any n ∈ ℕ. We show that A is an absolute 𝔾-ring for the Brown–McCoy radical 𝔾, if and only if A is in the radical class S determined by the unitary strongly prime rings. Moreover, A is an absolute nil ring if and only if A is an absolute J-ring, where J denotes the Jacobson radical.