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This work is motivated by the articles [9] and [19] in which the following two problems are solved. Let O be a finitely generated ℤ-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix s that is integral over O, describe all t such that O[s]= O[t]. (B) Fix s and t that are integral over O, describe all pairs (m, n) ϵ ℕ² such that O[sm]= O[tn]. In this article, we solve these problems and provide a uniform bound for a certain "discriminant form equation" that is closely related to Problem (A) when O has characteristic p > 0. While our general strategy roughly follows [9] and [19], many new delicate issues arise due to the presence of the Frobenius automorphism x → xp. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this article. [ABSTRACT FROM AUTHOR]

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue λF of a Mahler function F(z) and develop a quick test for the transcendence of F(z) over C(z), which is determined by the value of the eigenvalue λF. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of F(z). We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given. [ABSTRACT FROM AUTHOR]

Let A and B be two connected graded algebras finitely generated in degree one. If A is isomorphic to B as ungraded algebras, then they are also isomorphic to each other as graded algebras. [ABSTRACT FROM AUTHOR]

Let E be any directed graph, and K any field. We classify those graphs E for which the Leavitt path algebra LK(E) is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive. [ABSTRACT FROM AUTHOR]

Abstract: Lenagan and Smoktunowicz (2007) (see also Lenagan, Smoktunowicz and Young (in press) ) gave an example of a nil algebra of finite Gelfand–Kirillov dimension. Their construction requires a countable base field, however. We show that for any field k and any monotonically increasing function which grows super-polynomially but subexponentially there exists an infinite-dimensional finitely generated nil k-algebra whose growth is asymptotically bounded by . This construction gives the first examples of nil algebras of subexponential growth over uncountable fields. [Copyright &y& Elsevier]

Abstract: We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show that this result holds for the class of automaton algebras; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz. [Copyright &y& Elsevier]

We give a corrected and strengthened statement and proof of the ‘p-adic analytic arc lemma’ in a paper of the author (J. London Math. Soc. (2) 73 (2006) 367–379). We show that the analytic arc is guaranteed to exist for p ≥ 5 and give a counterexample showing that this sometimes cannot be done when p = 2. [ABSTRACT FROM PUBLISHER]

Abstract: We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite provided that the ring is either right noetherian (4.2) or countably generated over a large field (6.4). We give examples to show that, over countable fields, a just infinite algebra which is either affine or non-noetherian need not remain just infinite under extension of scalars. We also give a concrete classification of PI stably just infinite rings (5.5) and give two characterizations of non-PI stably just infinite rings in terms of Martindale''s extended center (3.4), (3.5). [Copyright &y& Elsevier]

Abstract: In the vein of Christol, Kamae, Mendès France and Rauzy, we consider the analogue of a problem of Mahler for rational functions in positive characteristic. To solve this question, we prove an extension of Cobham''s theorem for quasi-automatic functions and use the recent generalization of Christol''s theorem obtained by Kedlaya. [Copyright &y& Elsevier]

Abstract: We study prime algebras of quadratic growth. Our first result is that if A is a prime monomial algebra of quadratic growth then A has finitely many prime ideals P such that has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the non-zero prime ideals P such that has GK dimension 2 is non-zero, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non-zero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra A of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that has GK dimension 1. [Copyright &y& Elsevier]

Abstract: Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that is left Noetherian for any field extension K of k. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over k are finitely generated extensions of k. We give examples which show that need not remain left Noetherian if the hypotheses are weakened. [Copyright &y& Elsevier]

By modifying constructions of Beı˘dar and Small we prove that for countably generated prime F-algebras of finite GK dimension there exists an affinization having finite GK dimension. Using this result we show: for any field there exists a prime affine algebra of GK dimension two that is neither primitive nor PI; for any countable field F there exists a prime affine F-algebra of GK dimension three that has non-nil Jacobson radical; for any countable field F there exists an affine primitive F-algebra of GK dimension at most four with center equal to a polynomial ring; for a countable field F there exists a primitive affine Jacobson F-algebra of GK dimension three that does not satisfy the Nullstellensatz. [Copyright &y& Elsevier]