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Symmetric Elements in Free Powers of Rings
Author(s) -
Bergman G. M.,
Cohn P. M.
Publication year - 1969
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-1.1.525
Subject(s) - subring , ring (chemistry) , free product , mathematics , combinatorics , product (mathematics) , symmetric group , field (mathematics) , group ring , group (periodic table) , finite group , discrete mathematics , pure mathematics , physics , chemistry , geometry , organic chemistry , quantum mechanics
Let A be a ring over a (skew) field K and denote by P = A 〈 X 〉 the free product over K of a family of copies of a indexed by a set X . Any group G acting on X also acts on P , by permuting the factor rings. We determine the structure of the subring S of elements of P fixed by G , when G acts on X with finite orbits. Under mild assumptions on A (e.g. if A is augmented), S is the free product of copies of A and a ring with weak algorithm; the latter factor becomes a free algebra when K lies in the centre of A . When X is permuted semiregularly by G , the structure of P over S is also determined.