RINGS WHOSE FREE MODULES SATISFY THE ASCENDING CHAIN CONDITION ON SUBMODULES WITH A BOUNDED NUMBER OF GENERATORS

Let R be a ring such that every finitely generated free (respectively, every free) right R-module satisfies the ascending chain condition on n-generated submodules for every positive integer n; then any ring Mo rita equivalent to R has the same property. This is in contrast to rin gs R which satisfy the ascending chain condition on n-generated right ideals, for some fixed positive integer n, for in this case rings Mori ta equivalent to R need not have the same property. If R is a right an d left Ore domain and n is a positive integer such that the free right R-module R-R((n)) satisfies the ascending chain condition on n-genera ted submodules then so too does every free right R-module. Many exampl es are given of rings for which every finitely generated free (respect ively, every free) right module satisfies the ascending chain conditio n on n-generated submodules, for some positive integer n. (C) 1998 Els evier Science B.V.