Some Simple Properties of Simple Nil Rings

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

A Galois theory of commutative rings

Herstein's Lie theory revisited

Let A be an indiscrete locally compact primitive ring having minimal left ideals.Algebraically, A may be regarded as a dense ring of linear operators containing nonzero linear operators of finite rank on a vector space E over a division ring K, and furthermore if A is simple, then A contains only linear operators of finite rank.The topology of A induces topologies on F and A" in a natural way, so that K becomes a locally compact division ring and E a locally compact vector space over K. Theorems about locally compact division rings and locally compact vector spaces imply that if K is indiscrete, then E is necessarily finite dimensional over K, and thus A is the ring of all linear operators on a finite-dimensional vector space over a locally compact division ring.Kaplansky proved that if A has characteristic zero, then K is indeed indiscrete, but he showed by an example that E could be infinite dimensional.Kaplansky remarked [9, p. 459], however, that if A is simple, it seemed unlikely that F could be infinite dimensional, since otherwise "completeness appears to require the presence of linear transformations with infinitedimensional range."Our principal result in 1 is the verification of this conjecture; however, our argument is based not on the fact that a locally compact ring is complete, but rather on the fact that a locally compact space is a Baire space and on the symmetry between minimal left and right ideals in a simple ring.In 2 we shall present certain results about locally compact vector spaces over discrete division rings; these enable us to prove that a locally compact, indiscrete, central primitive algebra having minimal left ideals is finite dimensional provided its scalar field either is indiscrete, has characteristic zero, or is uncountably infinite.However, there is a very natural example of a locally compact, indiscrete, infinitedimensional central primitive algebra having minimal left ideals over a discrete countably infinite field of prime characteristic.1. Locally compact simple rings having minimal left ideals.We begin with a summary of the structure theory of a primitive ring A having minimal left ideals.Let e be a minimal idempotent of A, that is, an idempotent such that Ae is a minimal left ideal, or equivalently, such that eA is a minimal right ideal.Then eAe is a division ring, the additive group Ae becomes a right eAe-\tctor space by defining scalar multiplication to be the restriction of multiplication on A to

Structure Theory of Simple Rings Without Finiteness Assumptions

ATTENTION has been given in NATURE to various deductions from the results of Dr. P. E. Shaw's experiments on The Newtonian Constant of Gravitation as affected by Temperature (Phil. Trans., A, 544, 1916). So far as the present writer is aware, attention has not been directed to the suggestive remarks of the late Prof. G. F. Fitzgerald, to be found in his Helmholtz memorial lecture (Transactions Chemical Society, 1896, pp. 889–95). In the course of his reference to Helmholtz's contribution to the theory of vortex motion, Fitzgerald remarks:—It is difficult to weigh hot bodies accurately, and, in consequence, there does not seem to be any conclusive proof that the weight of a body does not change with its temperature. If it does not do so by a measurable amount, the simple vortex ring theory of matter can hardly be true.