Modules Over Rings of Words
William G. Leavitt · Proceedings of the American Mathematical Society · 1956
Introduction.A right module M over a ring K with unit is called a free module if it is finitely based.If the basis number of such a module is not invariant then it must necessarily contain an infinite independent set [l, Theorem 7].A free module may, on the other hand, contain infinite independent sets but nevertheless have invariant basis number.If, for example, K contains no zero divisors, then the ring K considered as a module over itself must have invariant basis number (1), since otherwise there would exist independent {a<} (i = l, • • • , n = 2) such that 1 = ^2" a,*,-.But the independence of the {a,} would then imply that #i