Vol. 97, No. 1, 1981

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Axiomatic radical and semisimple classes of rings

John Robert Fisher

Vol. 97 (1981), No. 1, 81–91

The results of this paper combine two areas of abstract mathematics: the theory of radicals for associative or alternative rings, and model theory for universal algebras. The main theorem provides necessary and sufficient conditions on the radical of any product ring and on the radical of any ultraproduct ring in order that the radical class and its corresponding semisimple class be finitely axiomatic (elementary). As a corollary, it follows that if a radical class of rings and its corresponding semisimple class are axiomatic, then they are both finitely axiomatic. In addition, several subsidiary results are given, and unanswered questions posed.

Mathematical Subject Classification 2000
Primary: 16A21, 16A21
Secondary: 03C52, 03C60, 08C10
Received: 30 May 1980
Revised: 9 March 1981
Published: 1 November 1981
John Robert Fisher