The module type of a ring

Abstract

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.