Rings with a finite set of nonnilpotents

[摘要] LetRbe a ring and letNdenote the set of nilpotent elements ofR. Letnbe a nonnegative integer. The ringRis called aθn-ring if the number of elements inRwhich are not inNis at mostn. The following theorem is proved: IfRis aθn-ring, thenRis nil orRis finite. Conversely, ifRis a nil ring or a finite ring, thenRis aθn-ring for somen. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.

[学科分类] 数学（综合）