Ideals in MOD-R and the {omega}-radical

Let R be an artin algebra, and let mod-R denote the categoryof finitely presented right R-modules. The radical rad = rad(mod-R)of this category and its finite powers play a major role inthe representation theory of R. The intersection of these finitepowers is denoted rad, and the nilpotence of this ideal hasbeen investigated, in [6, 13] for instance. In [17], arbitrarytransfinite powers, rad, of rad were defined and linked to theextent to which morphisms in mod-R may be factorised. In particular,it has been shown that if R is an artin algebra, then the transfiniteradical, rad, 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–Gabrieldimension of mod-R equals (that is, is undefined). More preciseresults on the index of nilpotence of rad for artin algebraswere proved in [14, 20, 24–26].