The Module Type of a Ring

Introduction. In the following we will consider only rings with unit. It is well known that a finitely based module (2) (over some specified ring) can have bases of different lengths (3). Examples of such modules are given in 1; 2; 3. A module will be said to have dimension n if it has a basis of length w, and if all of its bases are of the same length. A ring A will be called dimensional if all finitely based ^-modules have dimension. The class of dimensional rings includes all rings with the ascending chain condition on left ideals (left Noetherian rings) 4, p. 32, and thus all rings with the descending chain condition (left Artinian rings) 4, p. 71. The class also includes all commutative rings 5, p. 563 and all subrings (with the same unit) of a division ring 5, p. 563 (or, more generally, of an Artinian ring 6, p. 249). In this paper it is shown that a given ring can admit only certain characteristic types of finitely based modules, and rings are classified according to to their module type. There exists a natural ordering of such types, relative to which the type of the dimensional rings is maximal. The types are shown to form a lattice, with the lattice operations related to certain operations on rings of corresponding types. It is shown, further, that whenever there exists a unit-preserving homomorphism of A into A', then the type of A' is less than or equal to that of A. As a corollary, we have a result (due to O. T. O'Meara) according to which the dimensionality of A' implies that of A. This permits us to add substantially to the class of dimensional rings.