JOURNAL OF ALGEBRA 35, 308-341 (1975) Elements of Noncommutative Arithmetic I OSCAR GOLDMAN* U&vrsity of Pennsylvania, Philadelphia, Pennsylz~ania I91 74 arid University of California, Berkeley, Berkeley, Califor?lia 947211 Communicated by N. Jacobson Received January 15, 1974 INTRODUCTION The word “arithmetic” in commutative ring theory is generally used to describe that part of module theory of noetherian rings having some connec- tion with prime ideals. For our purposes, we shall retain the noetherian condition by restricting our considerations to rings which satisfy the ascending chain condition for left ideals. And since we do not believe that an arithmetic can be based on prime ideals (in noncommutative rings), we shall replace the term “prime ideal” by prime kernel functor as defined in [3]. That is, noncommutative arithmetic will mean the study of finitely generated modules over a left noetherian ring in relation to the primes of that ring. The simplest such relation is that of primary decomposition of modules. A reasonably satisfactory form of such a decomposition exists, for example, in [3]. Such a theory is qualitative, and until a quantitative theory is developed, arithmetic remains in a primitive state. Our purpose in this paper is to introduce the most basic and elementary definitions upon which a quantitative theory may be based. Much of the quantitative theory for commutative rings arises from the study of composition series in suitable modules using the length of such series to provide numerical invariants for the purpose at hand. For example, the general theory of ramification index of Auslander and Rim [l] is based on such lengths of modules. With commutative ring theory as a guide, it would seem reasonable to look to composition series also in the noncommu- tative case as a source of quantitative information. Let us stay with commutative rings a bit longer. If M is an R-module and * This research was supported by the r\iational Science Foundation. 308 Copyright .i, 1975 by I\cademic Press, Inc. All rights of reproduction in any form reserved.