# A reference book for Schur's lemma

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

• Why is it relevant that the algebra has a countable basis? – André Henriques Jun 25 '12 at 18:58
• @André: Take $A = M = \mathbb{C}(x)$. Every element is central, but only a few act as scalars! – Evan Jenkins Jun 25 '12 at 19:32
• I'd bet it must be also in Dixmier's Universal eneveloping algebras. – Vít Tuček Jun 27 '12 at 12:52

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

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.

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