Call a ring Λ (left) absolutelytorsion−f7⋅ee (ATF) if for every finite kernel functor σ (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 artinian 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.
