On Prime J-Rings with Uniform One-Sided Ideals

Abstract

We say that a ring S is a Jl-ring if it fulfills (J1). A Jr-ring is defined in an obvious way. A ring S is called a left quotient ring (in the sense of R. E. Johnson) of a subring R if every nonzero left R-submodule of S has a nonzero intersection with R. A right quotient ring is defined similarly. An extension ring S of a ring R is called a two-sided quotient ring of R if S is a left quotient ring of R, and also a right quotient ring of R. For any J1-rinig S there exists such a left quotient ring Si that every left quotient ring of S has an isomorphic, over S, image in SI. Si is unique up to isomorphism over S, and is called the maximal left quotienit ring of S. Similarly the maximal right quotient ring Sr exists for any Jr-ring S. In case S is a J-ring, it may be shown that S has the maximal two-sided quotient ring St. A ring S is called a continuous transformation ring if there is a pair of dual vector spaces (V, V') such that S is the ring of continuous linear transformations of the vector space V topologized by the V'-topology. This concept is right-left symmetrical; that is, S may be regarded as the ring of continuous linear transformations of the vector space V' topologized by the V-topology. (See [2; ?? 6, 7, Chap. IV]). A left ideal A of a ring S is called uniform if we have B n C ? 0 for any nonzero left ideals B and C such that B, C C A. A uniform right ideal is defined similarly. The main theorem of this paper is the following: Let S be a prime J-ring