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Recently, reading the book 'Skew Linear Groups' by M. Shirvani and B. A. F. Wehrfritz, I've encountered the following:

Let D be a division ring which is locally finite-dimensional over 
its centre but not finite dimensional over its centre.

Searching the web, I could not find the definition of being locally finite-dimensional.
The 'usual candidates' for a local property in rings are things to do with ideals, but as there are no non-trivial ideals in a division ring, the only thing that sounds remotely close is looking at finitely generated sub-algebras (when looking at $D$ as an algebra over it's centre) - is that correct? Can anyone point me to the definition of this?
Thanks in advance.

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  • $\begingroup$ Perhaps related to local subrings of $D$.. $\endgroup$ – Berci Feb 8 '13 at 20:59
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An algebra of any kind (Lie, associative, etc.) is locally finite-dimensional (also called locally finite) if every finitely generated subalgebra is finite dimensional over the ground field. See this.

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  • $\begingroup$ Thanks, that's exactly what I was looking for! $\endgroup$ – Dennis Gulko Mar 4 '13 at 9:12
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From what I've seen, a ``locally finite dimensional" representation $V$ of a Lie algebra $\mathfrak{g}$ is one for which $U(\mathfrak{g}) v$ is finite dimensional for every $v \in V$. Thus, perhaps in your case you should require $Z d$ to be finite-dimensional for every $d \in D$ where $Z$ is the center.

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