A companion to S.Lang's Algebra 4ed. - download pdf or read online

By Bergman G.M.

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Cf. 11 of the 245 notes). Notes on varying conventions. An author writing about categories has to make certain choices of convention. Lang has chosen to write morphism-sets as Mor(X, Y ); a more common notation is Hom(X, Y ) (even though one says ‘‘morphism’’ rather than ‘‘homomorphism’’). When one wants to specify the category unambiguously, one writes this morphism-set Hom (X, Y ); a simpler notation, also common, is (X, Y ). , some authors use one or another of these symbols to denote the set of morphisms from Y to X rather than the set of morphisms from X to Y !

12:8 in this Companion. ’’. (If we think of a free group on a set X as a group generated by an X-tuple of elements that satisfy no relations other than those implied by the group identities, and a group presented by generators and relations X and R as a group generated by an X-tuple of elements that satisfy no relations other than those implied by the relations in R and the group identities, we can understand why the latter was sometimes called ‘‘the free group with those generators and relations’’.

Such a group is isomorphic to Z ⁄ n Z, but not in a unique way. , for n = 4, the group of complex numbers {1, i, –1, –i }), so that it would be inconvenient to have the results stated only for the ‘‘standard’’ group. Note that if m is an exponent of the abelian group A, then any multiple rm of m is also an exponent of A. If we work with ‘‘standard’’ cyclic groups, we have a canonical embedding Z ⁄ m Z → Z ⁄ rm Z, taking [1] to [r], whose image is precisely the group of elements of Z ⁄ rm Z of exponent m.

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A companion to S.Lang's Algebra 4ed. by Bergman G.M.


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