Nilpotence of nil subrings implies more general nilpotence

Recently H]~RSTEIN and SMALL [2] have shown that in a ring R (associatiVe) satisfying various weakened forms of the ascending chain condition, every nil subri~g is nilpotent. In the attempt to extend these results to nil sub-semigroups of 17 a ~d to Lie or Jordan subrings of R, the author was led to the following general theorem: I/ every nil subring o/ a ring R is nilpotent, then so is every nil weakly closed subset o/ R. (A subset S of a ring R is weakly closed if it is closed under some binary op" eration  on R satisfying t  s = ts + ~(s, t) t + fl(s, t), where ~ (s, t) and fl (s, t) are A-linear combinations of positive powers of s, with ~ regarded as an algebra over a ring A of operators.)