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We want to find the reference book for some version of the schur's lemma which covers the following result

Let A be an assoiative algebra over {\mathbb C} with countable basis, then any central element acts on any simple A-module as a scalar.

Thanks

edited Jun 25 at 23:10

PaPiro
1,356●1●9●26

asked Jun 25 at 18:32

r_l
25●3

Why is it relevant that the algebra has a countable basis? – André Henriques Jun 25 at 18:58

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@André: Take $A = M = \mathbb{C}(x)$. Every element is central, but only a few act as scalars! – Evan Jenkins Jun 25 at 19:32

I'd bet it must be also in Dixmier's Universal eneveloping algebras. – robot Jun 27 at 12:52

Evan Jenkins · Accepted Answer · 2012-06-25 19:25:04Z UTC

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This is (an immediate consequence of) Lemma 2.1.3(b) in Chriss-Ginzburg.

answered Jun 25 at 19:25

Evan Jenkins
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Bugs Bunny · Answer 2 · 2012-06-27 08:08:47Z UTC

Doc, if you want to be anal with your references, you should quote Amitsur, A. S. Algebras over infinite fields. Proc. Amer. Math. Soc. 7 (1956), 35–48.

Otherwise, this is a well-known fact and you can just refer to it as "Amitsur's Trick" or "Noncommutative Nullstellensatz". Chapter 9 of McConnell-Robson-Noncommutative-Noetherian-Rings is devoted entirely to this property and its finer variations.

Johan Commelin · Answer 3 · 2012-06-25 19:02:39Z UTC

From Wikipedia:

David S. Dummit, Richard M. Foote. Abstract Algebra. 2nd ed., pg. 337.
Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0

From Google Books:

William Fulton, Joe Harris. Representation Theory.
William Arveson. An Invitation to C*-Algebras.

I hope that helps you a bit.

P.s: I wanted to post this in a comment, but the comment button isn't available for some reason.

Steven Spallone · Answer 4 · 2012-06-27 03:23:01Z UTC

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This is also in Bourbaki's Algebra 8 (most recent edition), Section 3, number 2, Example, page 43. This reference is online if you have access to SpringerLink.

answered Jun 27 at 3:23

Steven Spallone
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A reference book for Schur’s lemma

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