No abstract

We prove that for any infinite κ, the full symmetric group Sym(κ) is the product of at most 14 abelian subgroups. This is a strengthening of a recent result of M. Abért.

Abstract. Let C be the field of complex numbers and V a subvariety of (C-{0})".

ABSTRACT. Let R be an algebra over a field k, and P, Q be two nonzero finitely generated projective .R-modules. By adjoining further generators and relations to R, one can obtain an extension S of R having a universal isomorphism of modules, i: P®RS s Q <8)R S.We here study this and several similar constuctions, including (given a single finitely generated projective Ä-module P) the extension S of R having a universal idempotent module-endomorphism e: P ® S -► P ® S, and (given a positive integer n) the fc-algebra S with a universal /c-algebra homomorphism of R into its nXn matrix ring, /: R-► mn(S').A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring Rq (= k X k X k, k X k, and k respectively in the above cases), and hence to apply the theory of such coproducts.As in that theory, we find that the homological properties of the construction are extremely goorf: The global dimension of 5 is the same as that of R unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relationThese results allow one to construct a large number of unusual examples.We discuss the problem of obtaining similar results for some related constructions: the adjunction to R of a universal inverse to a given homomorphism of finitely generated projective modules, /: P -► Q, and the formation of the factor-ring R/Tp by the trace ideal of a given finitely generated projective R-module P (in other words, setting P = 0).The idea for a category-theoretic generalization of the ideas of the paper is also sketched.

No abstract