Hereditary Torsion Theories for Noetherian Rings

Abstract

For a commutative noetherian ring we have proved that every Gabriel topology ℑ arises from a set p of prime ideals as

$$ \\\mathfrak{F} = \{ a|a \not\subset \mathfrak{p},all\mathfrak{p} \in p\} ] $$

(Cor. VI.6.15). In the case of a non-commutative right noetherian ring the situation is not quite so simple. The main result of this chapter (Theorem 3.4) shows, however, that there is a large class of right noetherian rings for which the Gabriel topologies are uniquely determined by the prime ideals they contain. An important tool for obtaining this result is the non-commutative theory of associated prime ideals.