Rings whose free modules satisfy the ascending chain condition on submodules with a bounded number of generators Original Research Article

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 Morita equivalent to R has the same property. This is in contrast to rings R which satisfy the ascending chain condition on n-generated right ideals, for some fixed positive integer n, for in this case rings Morita equivalent to R need not have the same property. If R is a right and left Ore domain and n is a positive integer such that the free right R-moduleRR(n)) satisfies the ascending chain condition on n-generated submodules then so too does every free right R-module. Many examples are given of rings for which every finitely generated free (respectively, every free) right module satisfies the ascending chain condition on n-generated submodules, for some positive integer n.