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On finiteness of rings with finite maximal subrings. (English) Zbl 0798.16009

T. J. Laffey suggested that a ring with a finite maximal subring is finite. The authors prove this conjecture for \(PI\)-rings. Earlier the same result has been obtained for commutative rings by the first author and F. Guerriero [Int. J. Math. Math. Sci. 13, No. 3, 535-544 (1990; Zbl 0711.16011)] and for arbitrary rings by the second author [Commun. Algebra 21, No. 4, 1389-1392 (1993; Zbl 0793.16009)]. The example of A. Yu. Ol’shanskij [Dokl. Akad. Nauk SSSR 245, 785-787 (1979; Zbl 0431.20025)] of an infinite group all of whose proper subgroups are maximal of prime order, shows that the analogous conjecture for groups is false.

MSC:

16P10 Finite rings and finite-dimensional associative algebras
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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