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By Paula Ribenboim

ISBN-10: 0471718041

ISBN-13: 9780471718048

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Then, for x E A, we have x ( u § iv) - x - xu - x § i x v E I, so t h a t x u - x C I R e I , proving I is regular with u Re as a relative r. unity. The proof of the corresponding result when I is a r. ideal is similar. (v) Since ~7 is self-conjugate, by (iii), I-- I§ I-- Re i , I - A N ~ 7. By (iv), I is regular and by (ii), any relative r. unity for I is a relative r. unity for I is a relative r. unity for I. The bijectiveness of the m a p I ~ I + i I is immediate. 13. LEMMA. Let I be a regular I.

Real and Complex Algebras N for some y 0 E A . v z N whence iv is a relative r. unity for I. 14. Let A be a real algebra and A its complexification. If J" is a functional (= real-valued function) on A, the extension f of f given by" (x, y E A) / ( ~ + y) - f (~) + i f ( y ) is called the canonical extension of f (to A). It is straightforward to verify that f is a linear functional (respy. character) of A if f is a linear functional (respy. character) of A A character )~ of A is called real if X - )~1A is real-valued (and then X is a character of A).

But then, for any a E A, a - aaob c ker xo = M c L. (,) Since b E L and L is a 1. ideal we conclude from (,) t h a t a E L, which means L - A, proving M is maximal. 15. If A is an algebra and Il a 1. ideal of A, then the 42 Algebraic P r e l i m i n a r i e s quotient A # - A / I z is canonically a left A-module: a ( x § Ii) -- ax + Ii (a, x E A). 2). e. 1-1) if (It "A) - {0}. 1 . 5 . 1 6 . LEMMA. Let I -- Iz be a regular I. ideal of an algebra. Then: (i) The quotient module A # - A / I is cyclic.

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Algebraic numbers by Paula Ribenboim

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