| 摘 要： | Definition A subring A of a ring R is called a pseudo ideal, if for any r∈R and α∈A, r~2α∈A and αr~2∈A. The subring (R)~2={Σ±r_1~2…r_i~2|r_i∈R, t∈N} is called the square closed ring of R. Lemma The square closed ring of a division ring D is the intersection of all nontrivial pseudo ideals of D. Theorem Ⅰ If a simple ring R with a unit has a nontrivial pseudo ideal A containing an invertible element, R has characteristic p=2. In particular, the necessary condition for a division ring containing a nontrivial pseudo ideal is
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