Growth and uniqueness of rank
Let R be a ring. Then R is said to have UR (the unique rank property) if every free R-module has a well-defined rank. The main aim of the paper is to show that R has UR if R has a subring S of the type defined below and R is related to S in a way which generalizes being finitely-generated as an S-module. The ring S is monotone in the sense that if the direct sum of k copies of S embeds as an S-module in the direct sum of n copies of S, then k is less than or equal to n. In particular S could be commutative or a Goldie ring. The connection between R and S is given by a ``growth structure'' which we will not define precisely here, but it involves an increasing filtration of R by free S-modules of finite rank. It is shown that if the corresponding growth is subexponential then R is monotone and hence has UR.
scientific article; zbMATH DE number 4101398