Hereditary semisimple classes

W. G. Leavitt
1970 Glasgow Mathematical Journal  
It is well-known (see e.g. [1, p. 5]) that a class Jt of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class W if and only if it has the following properties: (a) UReJt, then every non-zero ideal /of R has a non-zero homomorphic image I/JeJt. In fact °UM is the radical class whose semisimple class is Jl. On the other hand, if 9> is a radical class, theny^1 = {Keif\ if / is a non-zero ideal of A", then !$&} is its semisimple class.
more » ... s semisimple class. If a class Jt is hereditary (that is, when ReJl, then all its ideals are in Jt), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2] ). The condition (satisfied in all associative or alternative classes) is that SfSP is hereditary for a radical class 0> if and only if 0>{t) £ &{R) for all ideals / of all rings Reir [3, Lemma 2, p. 595]. It is also well-known (see e.g. [1, pp. 6-7]) that, if Jt satisfies only condition (a), then°U Jl is a radical class such ihdXSffyJl is the unique minimal semisimple class containing Jl. However, for an arbitrary class Jt we have:
doi:10.1017/s0017089500000781 fatcat:ee3k4ksqcjfexlyy55ehcfr34y