Algebraic geometry over groups. I: Algebraic sets and ideal theory
type: Item
instance of: scholarly article
title: Algebraic geometry over groups. I: Algebraic sets and ideal theory
zbMATH Open document ID: 0938.20020
author: Gilbert Baumslag
author: Alexei Myasnikov
author: Vladimir Remeslennikov
published in: Journal of Algebra
publication date: 1999-12-20T00:00:00Z
review text: The authors present the first in the series of three papers with the object to lay the foundations of the theory of ideals and algebraic sets over groups. There is discovered a surprising similarity to elementary algebraic geometry -- hence its name. In the present paper they introduce group-theoretic counterparts to such notions as zero-divisors, prime ideals, the Lasker-Noether decomposition of ideals as intersections of prime ideals, the Noetherian condition, irreducibility, and the Nullstellensatz. Some new concepts arise that are interesting for the group theory. The main of them is the notion of a \(G\)-group, where \(G\) is a fixed group. This group \(G\) plays the role of the coefficient ring. A group \(H\) is called a \(G\)-group if \(H\) contains a designated copy of \(G\). Such groups \(G\) form a category with naturally defined \(G\)-morphisms. The kernels of morphisms are called ideals. One can talk in a natural way about free \(G\)-groups, finitely generated and finitely presented \(G\)-groups and so on. In particular, the finitely generated free \(G\)-groups are the free products of \(G\) with the free groups of finite ranks. So, we can consider such a group as a non-commutative analogue of a polynomial algebra over a unitary commutative ring in finitely many variables. Elementary properties of algebraic sets and the Zariski topology are developed. The important notion of coordinate group is introduced. Equivalence of the categories of affine algebraic sets and coordinate groups is explained. Some decomposition theorems are proved.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20J15
zbMATH DE Number: 1348088
zbMATH Keywords: algebraic geometry
zbMATH Keywords: algebraic sets over groups
zbMATH Keywords: affine geometry
zbMATH Keywords: \(G\)-groups
zbMATH Keywords: separation
zbMATH Keywords: discrimination
zbMATH Keywords: Nullstellensatz
zbMATH Keywords: representations
zbMATH Keywords: quasivarieties
zbMATH Keywords: equationally Noetherian groups
zbMATH Keywords: zero-divisors
zbMATH Keywords: prime ideals
zbMATH Keywords: categories of groups
zbMATH Keywords: free groups
zbMATH Keywords: free products
zbMATH Keywords: coordinate groups
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
full work available at URL: jabr.1999.7881
OpenAlex ID: W2061741468
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DOI: 10.1006/JABR.1999.7881
description: scientific article; zbMATH DE number 1348088