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On (σ, τ)‐Derivations in Free Power Series Rings
Author(s) -
de Jooste T. W.
Publication year - 1980
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-40.1.53
Subject(s) - mathematics , endomorphism , automorphism , kernel (algebra) , filtration (mathematics) , formal power series , power series , ring (chemistry) , pure mathematics , commutative ring , inverse , series (stratigraphy) , commutative property , discrete mathematics , mathematical analysis , geometry , paleontology , chemistry , organic chemistry , biology
If K is a commutative ring and R an inversely filtered K ‐algebra which satisfies the inverse weak algorithm, then the kernel of a (σ, τ)‐derivation in R (where σ is any automorphism and τ any endomorphism of R ) is a semifir, and R itself is a flat left module over the kernel. Consequently, the fixed ring of any endomorphism of R is still a semifir. In the special case where R is a free power series ring over a field of characteristic zero these (σ, τ)‐derivations are continuous maps in the filtration topology. There also exists a fairly extensive class of continuous (σ, τ)‐derivations whose kernels are again power series rings because they satisfy the inverse weak algorithm with respect to the induced filtration. This is not true for all derivations in R .