Let R be an infinite ring with a maximal finite subring. We prove that R has a largest finite ideal and a largest finite nilpotent ideal N. By a B-ring, we mean an infinite ring with 1 containing a maximal finite subring which is a subfield containing 1. It is shown that R/N≌V V , where U is a finite ring, V is a finite direct sum of matrix rings over B-rings, and W is a ring containing no nonzero finite subrings.
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