 
Summary: ISRAELJOURNALOF MATHEMATICS,Vol. 64, No. 2, 1988
GROWTH AND UNIQUENESS OF RANK
BY
ELI ALJADEFF AND SHMUEL ROSSET
School ofMathematical Sciences, Raymond and BeverlySackler Faculty of Exact Sciences,
Tel Aviv University, Tel Aviv 69978, Israel
ABSTRACT
We prove that algebras of subexponential growth and, more generally, rings
with a subexponential "growth structure" have the unique rank property. In
the opposite direction the proof shows that if the rank is not unique one gets
lowerbounds on the exponent ofgrowth. Fixing the growthexponent it shows
that an isomorphism between free modules of greatly differing ranks can only
be implemented by matrices with entries of logarithmically proportional high
degrees.
It is well known that there exist rings over which the rank of a free left
module is not uniquely defined, i.e., they have modules that are free on two
bases of different cardinalities. A ring for which this distressing phenomenon
does not happen is said to have the (left) uniquerankproperty;we also say that
it has the "UR" property or, simply, that it "has UR". A commutative ring
always has the UR property since it has a nontrivial (i.e., with 1 going to 1)
