Some Remarks on ^-transitive Rings and Linear Compactness

  • KENNETH G. WOLFSON
  • Published 2010

Abstract

Johnson in [3] has introduced the concept of a p-transitive ring which generalizes the notion of a dense ring of linear transformations. We give necessary and sufficient conditions that an abstract ring be isomorphic to a ^-transitive ring which contains finite-valued linear transformations. The condition (2) used here is a modification of one used by Baer [l ] in his characterization of the endomorphism ring of a primary Abelian operator group. This condition is also related to the linear compactness of a ring considered as a right module over itself. This enables one to conclude that a primitive ring with minimal ideals which is linearly compact (in any topology in which it is a topological ring) is the ring of all linear transformations of a vector space, and that a primitive Banach algebra is linearly compact only when it is finite dimensional. A ring £(P, A) of linear transformations of the vector space A over the division ring F is called ^-transitive if to every set of less than N„ elements ay of A, linearly independent over F, and any set of elements bj of A, in one-one correspondence with the a,-, there exists a transformation <r in EiF, A), such that aja = bj for all/ Let K be an abstract ring and P an arbitrary subset thereof. The right ideal of all elements k in K which satisfies Pk = 0 shall be called a right annulet. Now let W= WiK) be the class of all right annulets which are cross-cuts of a finite number of maximal right annulets of K. By a PF-coset is meant a coset of an ideal in the set WiK). If EiF, A) is any ring of linear transformations and Sis a subspace of A, we denote by RiS) the totality of transformations ff£.E(P, A) satisfying 5<r = 0.

Cite this paper

@inproceedings{WOLFSON2010SomeRO, title={Some Remarks on ^-transitive Rings and Linear Compactness}, author={KENNETH G. WOLFSON}, year={2010} }