Free and Residually Artinian Regular Rings

A new measure of growth for countable-dimensional algebras. I

“…Here our aim is to establish the following, somewhat surprising, linear growth result which improves the Goodearl, Menal, and Moncasi embedding [GMM,Proposition 2.1] mentioned in the Introduction. Theorem 2.1.…”

“…Must A be unit-regular? The latter is not true if we require only that A has zero bandwidth dimension because it is known [GMM,Corollary 2.3] that the free regular algebra on a countable set over a countable field F embeds infill Mi(F). Incidentally, it was for the proof of this result that Goodearl, Menal, and Moncasi needed the fact that countable-dimensional algebras embed in BiF).…”

“…One reason for this, perhaps, is that the nice "arithmetic" functions provided by a finite matrix representation-such as trace, determinant, rank, etc.-would appear to have no cousins in the infinitedimensional case. However a recent and surprising result by Goodearl, Menal, and Moncasi [GMM,Proposition 2.1] offers fresh hope for infinite matrix representations of countable-dimensional algebras A over a field F: it says that such A can be embedded in the algebra 5(F) of all co x co matrices over F which are simultaneously row-finite and column-finite. (Note: cox co matrices are just No x No matrices with their rows and columns ordered in the standard way.)…”

Row and column finite matrices

“…Recently it was shown that any countably generated algebra over a field K can be embedded in RCFM(K) [3,Proposition 2.1]. This result, in turn, was used to define a new dimension function on such algebras [5].…”

A new measure of growth for countable-dimensional algebras

“…Its dimension values for finitely generated algebras exactly fill the unit interval [0,1 ]. Since the free algebra on two generators turns out to have dimension 0 (although conceivably some Noetherian algebras might have positive dimension!…”