Absolutely torsion-free rings

Abstract

Call a ring Λ (left) absolutely torsion-free (ATF) if for every finite kernel functor a (i.e., a topologizing filter of nonzero left ideals), σ (Λ) = 0. Since a commutative ring is ATF iff it is an integral domain, ATF rings may be viewed as generalizations of domains. Now an ATF ring is a prime ring, but there are even primitive rings that are not ATF. However if Λ is either finite as a module over its center, or finite dimensional and nonsingular as a left Λ-module, then Λ is ATF iff it is prime—in which case Λ is right ATF as well. The class of ATF rings is closed under the formation of polynomial rings, overrings in the maximal quotient ring, and Morita equivalence, but not under subrings. If Λ is ATF with maximal left quotient ring Q, then Q is simple, selfinjective and von Neumann regular. Furthermore Q is arti- nian iff Λ is (left) finite dimensional. An interesting class of ATF rings are the hereditary noetherian prime rings (HNP). Techniques used in deriving properties of ATF rings show that every ring between an HNP ring Λ and its maximal quotient ring is itself a ring of quotients of Λ with respect to some idempotent kernel functor, and thus is HNP itself. © 1973 Pacific Journal of Mathematics.