Alfred Goldie

Artinian quotient rings of ideal invariant noetherian rings

A ring R is ideal invariant on the right if for any two ideals 5’ and T the right Krull dimension ( SITS JR of the bimodule SITS is less than or equal to the right Krull dimension of R/T. This condition was introduced by Warfieldl, it is a weaker form of the concept of right ideal invariance due to Stafford [15] which requires the above inequality to hold whenever T is a right ideal. In recent years, several conditions have emerged which assure the existence of artinian classical quotient rings for various types of (left and right) noetherian rings. One of these is K-homogeneity, a condition which requires the Krull dimension 1 A jR of each non-zero right ideal A of the ring R to be equal to j R lR . The most general theorem in this connection appears in [8], it establishes the existence of a right artinian quotient ring Q(R) for any K-homogeneous ring R with right Krull dimension whose nil radical N is weakly right ideal invariant, meaning that N satisfies ) N/AN IR < ) R/N IR for any right ideal AwithjR/Aj<jR/NI,. Two previously known theorems are special cases of this: Gordon’s result [5] for FBN rings and Lenagan’s [lo] for noetherian rings with Krull dimension one. The assumption of R being K-homogeneous is by no means necessary for the existence of an artinian quotient ring. In general, the largest ideal with Krull dimension smaller than that of R does not even have to be a direct summand.

Strongly regular elements of noetherian rings

Any regular element of a commutative ring remains regular modulo every annihilator ideal of the ring. In general, this is not the case if the assumption of commutativity is dropped. Although certain elements, for example the units, stay regular modulo every right or left annihilator ideal, this may not be the case for other regular elements. Thus the multiplicatively closed set Y(O) of all those regular elements which are well behaved in this sense deserves some special attention, its elements are called strolzgly regular. This set is particularly interesting for questions concerning quotient rings, for it turns out that in a right noetherian ring any regular right Ore set is contained in Y(O) (Proposition 1.6). Thus the equality of -i”(O) and g(O), the set of regular elements, becomes a necessary condition for the existence of a full right quotient ring. For a (left and right) noetherian ring R it turns out that Y(O) = n g(P), where P runs through the set of all prime middle annihilator ideals (Theorem 2.1). A similar description of G?(O) in terms of finitely many affiliated primes obtained by L. Small and J. T. Stafford [8] thus makes the comparison of the two sets relatively easy. Although the inequality of Y(O) and @Y(O) provides a criterion for ruling out a full right quotient ring, equality of the two sets by no means guarantees its existence (Example 2.6).

A study in Krull dimension

The classica KrulI dimension of a ring, defined in terms of Iengths of chains of prime ideals, extends naturally enough to non-commutative rings and has a certain utility in that area. However it is not a sensitive indicator of the structure of many rings, simple rings being the extreme example. In the paper of P. Gabriel and R. Rentschler [3] a new Krull dimension was introduced with advantages over the old definition. It applies to modules as well as to rings and, in the extreme case of simple rings, gives an indication of how far the ring deviates from being artinian simple. A detailed and careful account of the elementary properties of this dimension, together with certain applications, will be found in G. Krause [6]. We shall assume that the elementary properties are known to the reader and study the practical effects which the presence of Krull dimension imparts to the ring structure. Since rings with minimum condition are precisely those with zero Krull dimension (in the new sense), they serve as a starting point for inductive arguments in the general case and also as an indication of results which are to be hoped for. It has been shown by R. Gordon and C. Robson (unpublished) that a semi-prime ring with Krull dimension is a Goldie ring and we give here a short proof of this useful theorem. Using this theorem we prove that the classical Krull dimension is less than or equal to Krull dimension. This was proved in Krause [6] for left noetherian rings. Our main theorem states that the nil radical is nilpotent in a commutative ring with finite KruIl dimension. Such rings need not be noetherian and an example is given of a ring in which both Krull dimensions are unity but the ring does not have the

Algebras which represent their linear functionals

This paper was originally intended to contain a generalization of a theorem of Banach on the extension of linear functionals. This generalized theorem now appears as a by-product of a study of a class of algebras which we believe to be of much greater interest than the theorem itself. Let X be a vector space over the real field and let π( x ) be a sub-additive, positive-homogeneous functional on X. Banach (( 2 ), pp. 27–9) proves that any real linear functional f on a subspace X 0 of X which satisfies f ( x ) ≤ π( x ) on X 0 can be extended to a real linear functional F on X with F ( x ) ≤ π( x ) on X. One of the essential differences between this theorem and the Hahn-Banach theorem is that π can take negative values.