Introduction

Unlike Chapters 2, 3, 4, and 5, in the next few chapters the ring need not have an identity, (i) Once an adequate notation and terminology is established, it is just as easy to prove everything for a general ring, (ii) If all our theorems were proved only for rings with an identity, they could not be applied to subrings or ideals or one sided ideals of a ring. However, some of the theorems proved here are useful tools for proving other theorems, for example by applying them to various subrings of a ring when such arise naturally in the course of some proof, (iii) Many interesting rings that occur naturally cannot have an identity as a matter of principle, e.g. nilrings, or rings of linear transformations on an infinite dimensional vector space having finite dimensional images, (iv) Some of the more influential previous authors have always traditionally treated these topics for rings without an identity, (v) Modules of the form R/L where L is a right ideal occur frequently in ring theory. To insist that R have an identity forces R/L to be cyclic, R/L = (1 + L)R, and the objects of study are trivialized. In the case when R is the ring of integers, R/L becomes a cyclic abelian group. To insist that the ring R should have an identity in general is the same as for an abelian group theorist to restrict his attention only to cyclic abelian groups.