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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

The Formanek polynomial gives an explicit expression for the calculation of central elements in a prime ring with polynomial identity, in such a form as to provide a set of linear maps of the ring into its centre. This note makes use of properties of these maps with the object of giving a direct proof of the remarkable theorem of M. Artin on the relationship between Azumaya algebras and rings with polynomial identities. We show that such a central map can be chosen to have some of the crucial properties of the reduced trace in matrices, notably that of non-degeneracy, and our proof of the theorem rests on these properties. The method can also be used for the study of prime PI rings in general; this is illustrated by giving new proofs of theorems of Formanek and Cauchon. The changes of proof essentially amount to the elimination of the need for a change of rings to obtain a split extension in simple algebras. The main theorem is the following.

1. Let R be a ring and M a right R-module. The Krull dimension [M [ of M is defined by induction. When M = 0, set I M ] = 1 and for an ordinal ~, knowing that I M I * ~, then set I MI= = provided that there is no infinite descending chain M = Mo = M1 = M2 = "" of submodules such that [ M i_ 1/M,I "~ ~. There need be no such ordinal ct, indeed its existence for M is an indicator of deviation from artinian modules, since M has minimum condition for submodules if and only if lM I = o. The Krull dimension of a ring R is defined to be [R R [. Strictly speaking, this is the rioht Krull dimension of R and, equally well, one can consider the left Krull dimension of R which may be different. For instance, there exist principal right ideal domains (right Krull dimension one) which are not left Ore, hence violate (A.05) and so do not have left Krull dimension. The idea originated in Gabriel-Rentschler [A2]. Some elementary properties are readily established (proofs in Gordon-Robson [A8]). Let 0 ~ K ~ M ~ N ~ 0 be exact; then: (A.01) IM 1 -suP(IN I, IKI>.

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 X0 of X which satisfies f(x) ≤ π(x) on X0 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.

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).

course that (i,,(c))* is an integral domain. Finally, since (eS (c))* is a finite )dule over the complete local domain(,,()* and has no zero-divisors, it must a complete local domain.4 This completes the proof of our theorem. We shall conclude this note by a few remarks on the hypothesis of our theorem. observe that in our formulation of the local Theorem of Bertini, we have ;roduced the subring e and a system of regular parameters x,, ..., xt in , lile in the final assertion of the theorem only the three elements xi, x2, X3 enter ;o the picture. The question naturally arises whether it is necessary at all to ;roduce the regular local domain ( or even the system of parameters xl, ..., in t. As to the latter, one finds a measure of justification in the fact that, en in the equal-characteristic case where T9 contains a coefficient field k, an 9itrary set of three analytically independent elements xl, x2, X3 may generate an ideal of rank less than 3 and hence cannot be extended into a system of rameters in ?. This is a well-known situation where a local ring may contain as ;ubring a local ring of higher dimension, and one can easily obtain examples to ow that the local Theorem of Bertini cannot hold for an arbitrary set of three

In the paper published in Mathematical Proceedings 79, 393–399, there is a gap in the proof of Theorem 2 which may present a difficulty to the reader. It occurs in the proof of non-degeneracy of the map g in the ring of matrices Zn, where Z is a finite field. Matrices are chosen, the first is diagonal with distinct entries and the others areWhen n is larger than the number of elements in Z, then such an x0 does not exist in Zn. The argument can, however, be sustained in the following way.