Hereditary semisimple classes

Leavitt, W. G.

Glasgow mathematical journal, Tome 11 (1970) no. 1, pp. 7-8

Résumé

It is well-known (see e.g. [1, p. 5]) that a class M of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:(a)if M, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈M.(b) If R∈ but R∉M, then R has a non-zero ideal I∈, where M = {K ∈ | every non-zero K/H∉M}. In fact M is the radical class whose semisimple class is M. On the other hand, if ℘ is a radical class, then I℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class M is hereditary (that is, when R∈M, then all its ideals are in M), 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 ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

Détail

DOI : 10.1017/S0017089500000781

Bibliographie

[1] 1.Divinsky, N. J., Rings and Radicals (Toronto, 1965). Google Scholar

[2] 2.Leavitt, W. G. and Armendariz, E. P., Non-hereditary semisimple classes, Proc. Amer. Math.Soc. 18 (1967), 1114–1117. Google Scholar | DOI

[3] 3.Anderson, T., Divinsky, N., and Sulińnski, A., Hereditary radicals in associative and alternative rings, Canad. J. Math. 17 (1965), 594–603. Google Scholar | DOI

[4] 4.Rjabuhin, Ju. M., Lower radicals of rings, Mat. Zametki 2 (1967), 239–244. Google Scholar