Equational Noetherian Algebras and Chain Conditions

TL;DR

This paper explores the connection between equational noetherian properties and chain conditions in algebras, providing a Hilbert basis theorem formulation and a criterion for varieties.

Contribution

It introduces a new criterion for assessing the Hilbert basis property in algebraic varieties and clarifies the relationship between noetherianity and chain conditions.

Findings

Established a criterion for the Hilbert basis property in varieties

Linked equational noetherianity with ascending chain conditions

Provided a generalized formulation of Hilbert's basis theorem

Abstract

In this article, we describe the relation between the properties of being equational noetherian and ascending chain condition on ideals of an arbitrary algebra. We also give a formulation of Hilbert's basis theorem for varieties of algebras and obtain a criterion to investigate it for a given variety.