IDEALS IN MOD-$R$ AND THE $\omega$-RADICAL

Abstract

Let be an artin algebra, and let mod- denote the category of finitely presented right -modules. The radical of this category and its finite powers play a major role in the representation theory of . The intersection of these finite powers is denoted , and the nilpotence of this ideal has been investigated, in , for instance. In , arbitrary transfinite powers, , of rad were defined and linked to the extent to which morphisms in may be factorised. In particular, it has been shown that if is an artin algebra, then the transfinite radical, , the intersection of all ordinal powers of rad, is non-zero if and only if there is a 'factorisable system' of morphisms in rad and, in that case, the Krull-Gabriel dimension of equals (that is, is undefined). More precise results on the index of nilpotence of rad for artin algebras were proved in.