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A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.

In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If Rk≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [32], [33]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8]., Comment: arXiv admin note: text overlap with arXiv:1207.6503

For an arbitrary countable field, we construct an associative algebra that is graded, generated by two elements is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl. The Jacobson radical is very important for the study of noncommutative algebras. For a given ring R one usually studies the Jacobson radical J(R) of R, and the semiprimitive part R/J(R). As related evidence of a connection between these notions, a result of Amitsur says that the Jacobson radical of a finitely generated algebra over an uncountable field is nil, and it is known that all nil rings are Jacobson radical. It is important to know when Jacobson radical are nil because nil rings have interesting properties. For example subalgebras of nil algebras are nil, which does not hold in general for Jacobson radical rings. The Jacobson radical is important for determining the structure of rings and is a generalization of the Wedderburn radical for finitely dimensional algebras.

Let K be a field, and let R=⊕n∈NRn be a finitely generated, graded K-algebra which is a domain. It is shown that R cannot have Gelfand-Kirillov dimension strictly between 2 and 3.

In [1] Amitsur conjectured that if a polynomial ring in one indeterminate is Jacobson radical then it is a nil ring. We shall construct an example disproving this conjecture.

We describe the nil radical of power series rings in non-commuting indeterminates by showing that a series belongs to the radical if and only if the ideal generated by its coefficients is nilpotent. We also show thatt the principal ideals generated by elements of the nil radical of the power series ring in one indeterminate are nil of bounded index.