| Summary: | 博士 === 國立臺灣大學 === 數學研究所 === 83 === We prove that if R is a prime ring with a s-derivation d
such that sd=ds and the image of d is contained in two of the
three nuclei then R is associative or d^2=2d=0. This theorem
generalizes Suh''s result [16] and characterizes the prime
associative rings of characteristic not two by a nonzero skew
derivation. Thus we obtain Posner''s theorem [14] for the prime
nonassociative rings of characteristic not two. We also prove
that if R is a prime ring with a s-derivation d such that sd=ds
and d^n(R) .lhkeq. G where n is a fixed positive integer, then
R is associative or d^{3n-1}=0. We apply this result to improve
and extend one of Jensen''s results [8] for the non-associative
prime rings. We pose a general problem which tries to
characterize the prime associative rings by algebraic skew
derivations and a special case of this problem is proved. If
R is a prime ring with a skew derivation d such that ax+d(x)
.in. G for all x in R where a .in. Z or a .in. G .intersection.
C, then R is associative or d(G)=0 which implies d(ax)+d^2(x)=0
for all x in R. We show that if R is a semiprime ring and all
the associators is contained in two of the three nuclei then
the three nuclei are equal and 2(R,R,R)=0. Hence Kleinfeld''s
result [10] has improved. We also prove that if R is a simple
ring and satisfies (R,R,R) .lhkeq. N and [[R,R],R] .lhkeq. N
then N=M=L and 2(R,R,R)=0. In the course of the proof of this
result, we extend the theorem of [21]. In the joint paper,
Hentzel and Yen [7] generalize the result of [21] for the
semiprime ring case. Moreover, Hentzel proves that if R is
of characteristic not two, i.e. 2x=0 .darrr. x=0 in R, then
(R,R,R)^3=0 for the general ring case.
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