Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)
ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1)).
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Suppose is a Lie ring and is an integer. The map is an endomorphism of as a Lie ring if and only if the derived subring of has exponent dividing .