Related papers: On Bergman's Diamond Lemma for Ring Theory

The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of…

This paper gives a generic form of the diamond lemma, which includes support for additive and topological structures of the base set, and which does not require any further structure (e.g. an associative multiplication operation) to be…

This is the original version of my Ph.D. thesis. The main results have been divided up between papers arXiv:2111.09784 and arXiv:2204.10924. This paper has been kept on the arXiv to preserve some proofs of elementary lemmas that will be…

We study diamond-alpha integrals on time scales. A diamond-alpha version of Fermat's theorem for stationary points is also proved, as well as Rolle's, Lagrange's, and Cauchy's mean value theorems on time scales.

In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke…

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…

In this note, the correction to the proof of one theorem in some our previous paper [arXiv:1302.0589] will be given.

We give a new proof of a lemma by L. Shepp, that was used in connection to random coverings of a circle.

The purpose of this article is to provide a version of Bloch-Wigner theorem over the class of rings with many units.

We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature.

Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach…

We propose a proof of the Lagrange Interpolation Formula based on the Chinese Remainder Theorem for arbitrary rings. Even such relationships are known, we think that our viewpoint is worth being published.

The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian groups, eliminating unnecessary additional hypotheses used in earlier versions.

We close a gap appearing at the same time in the author's thesis "Iterated rings of bounded elements and generalizations of Schm\"udgen's theorem" [1] and in the author's article "Iterated rings of bounded elements and generalizations of…

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor $Nr$ from the category of special $\ld$-rings into the category of special…

This is an exposition, in 12 pages including all prerequisites and a generalization, of Karamata's little known elementary proof of the Landau-Ingham Tauberian theorem, a result in real analysis from which the Prime Number Theorem follows…

In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.

This is an old article of 2000. Its aim is to illustrate how a Lie-theoretic result of Zelmanov enables one to treat various problems in group theory.

The purpose of this paper is to prove that certain limits of polynomial rings are themselves polynomial rings, and show how this observation can be used to deduce some interesting results in commutative algebra. In particular, we give two…

This article is about the mathematics of ringing the changes. We describe the mathematics which arises from a real-world activity, that of ringing the changes on bells. We present Rankin's solution of one of the famous old problems in the…