Algebraic geometry over groups. I: Algebraic sets and ideal theory.

*(English)*Zbl 0938.20020The 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.

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.

Reviewer: V.A.Roman’kov (Omsk)

##### MSC:

20E05 | Free nonabelian groups |

20E34 | General structure theorems for groups |

14A22 | Noncommutative algebraic geometry |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20E07 | Subgroup theorems; subgroup growth |

20F65 | Geometric group theory |

20J15 | Category of groups |

##### Keywords:

algebraic geometry; algebraic sets over groups; affine geometry; \(G\)-groups; separation; discrimination; Nullstellensatz; representations; quasivarieties; equationally Noetherian groups; zero-divisors; prime ideals; categories of groups; free groups; free products; coordinate groups
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\textit{G. Baumslag} et al., J. Algebra 219, No. 1, 16--79 (1999; Zbl 0938.20020)

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