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2017 | OriginalPaper | Buchkapitel

1. First Principles

verfasst von : Gene Freudenburg

Erschienen in: Algebraic Theory of Locally Nilpotent Derivations

Verlag: Springer Berlin Heidelberg

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Abstract

Throughout this chapter, assume thatB is an integral domain containing a fieldk of characteristic zero.B is referred to as ak-domain.B denotes the group of units ofB and frac(B) denotes the field of fractions ofB. Further, Aut(B) denotes the group of ring automorphisms ofB, and Aut k (B) denotes the group of automorphisms ofB as ak-algebra. IfAB is a subring, then tr.deg A B denotes the transcendence degree of frac(B) over frac(A). The ideal generated byx 1, , x n B is denoted by either (x 1, , x n ) orx 1 B + ⋯ +x n B. The ring ofm ×n matrices with entries inB is indicated by \(\mathcal{M}_{m\times n}(B)\) and the ring ofn ×n matrices with entries inB is indicated by \(\mathcal{M}_{n}(B)\). The transpose of a matrixM isM T .

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Fußnoten
1
The termplinth commonly refers to the base of a column or statue.
 
2
Note that the termregular action is also used in the literature to indicate an algebraic action. A regular action is thus distinguished from arational action , which refers to the action of a group by birational automorphisms.
 
3
Vasconcelos’s definition of “locally finite” is the same as the present definition of locally nilpotent. Apparently, the terminology at the time (1969) was not yet settled.
 
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Metadaten
Titel
First Principles
verfasst von
Gene Freudenburg
Copyright-Jahr
2017
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-55350-3_1