Some remarks on 𝜈-transitive rings and linear compactness

Abstract

Johnson in [3] has introduced the concept of a v-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 v-transitive ring which contains finite-valued linear transformations. The condition (2) used here is a modification of one used by Baer [I ] 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 E(F, A) of linear transformations of the vector space A over the division ring F is called i-transitive if to every set of less than K, elements aj of A, linearly independent over F, and any set of elements bj of A, in one-one correspondence with the aj, there exists a transformation oin E(F, A), such that ajo = bj for all j. 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= W(K) be the class of all right annulets which are cross-cuts of a finite number of maximal right annulets of K. By a W-coset is meant a coset of an ideal in the set W(K). If E(F, A) is any ring of linear transformations and S is a subspace of A, we denote by R(S) the totality of transformations oCE(F, A) satisfying So=0.