Rings With Minimal Condition for Left Ideals

Abstract

One finds in the literature thorough-going discussion of rings without radical and with minimal condition for ideals (semi-simple rings). For the structure of ring whose quotient-ring with respect to the radical is semi-simple one can refer to the investigations of K6the (see K below). In this paper we shall examine the structure of ring with radical R D 0 and with minimal condition for ideals (general MLI ring). The key-stone of our investigations is the fact that the radical of is nilpotent, and this result we shall establish in ?1. In ?2 we shall prove that the sum of all minimal non-zero ideals is completely-reducible ideal 9A, and in ?3 we shall examine the distribution of idempotent and nilpotent ideals in 9)1. In ??4-6 we shall discuss the two extreme cases: (1) when is nilpotent, and (2) when is idempotent. For non-nilpotent we shall prove that the existence of either right-hand or left-hand identity is sufficient for the existence of composition series of ideals of A. If is any MLI ring, one can find smallest exponent k such that Ak = Ak+l = ... . In ?7 we show that is the sum of Ak (which is idempotent) and nilpotent MLI We wish to emphasize the fact that is to be regarded throughout as ring without operators. In ?8, however, we shall see that some of our most interesting results are valid for operator domains of certain type. We conclude the Introduction with an explanation of our notation and terminology. Rings and subrings will usually be denoted by roman capitals; we shall use gothic letters when it is desirable to emphasize the fact that subring is an ideal. By the statement a is (right) ideal of A we shall mean that is an additive abelian group which admits the elements of as left-hand (right-hand) operators. Observe that our definition does not' imply that is subring of A. The term left ideal, with no qualifying phrase, will always mean left ideal of the basic ring. ring with minimal condition for (right) ideals which are contained in itself will be called an MLI (MRI) Finally we point out that if and b are subrings of A, then [a, b] denotes the cross-cut of and b, while (a, b) represents the compound (join) of and b-i.e.