Centralizers on prime and semiprime rings
Centralizers on prime and semiprime rings
Centralizers on prime and semiprime rings
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<str<strong>on</strong>g>Centralizers</str<strong>on</strong>g> <strong>on</strong> <strong>prime</strong> <strong>and</strong> semi<strong>prime</strong> <strong>rings</strong> 235<br />
Thus we have the relati<strong>on</strong> S(x)y[S(x);T(x)] + S(x) 2 [y; T(x)]+<br />
S(x)T (x)[S(x);y] = 0 which can be written in the form S(x)y[S(x);T(x)] +<br />
S(x) 2 yT(x) , S(x)T (x)yS(x)+S(x)[T (x);S(x)]y = 0 whence it follows<br />
(15) S(x)y[S(x);T(x)] + S(x) 2 yT(x) , S(x)T (x)yS(x) =0<br />
according to (12). Left multiplicati<strong>on</strong> of (15) by T (x) gives<br />
(16) T (x)S(x)y[S(x);T(x)] + T (x)S(x) 2 yT(x) , T (x)S(x)T (x)yS(x) =0:<br />
The substituti<strong>on</strong> T (x)y for y in (15) gives<br />
(17) S(x)T (x)y[S(x);T(x)] + S(x) 2 T (x)yT(x) , S(x)T (x) 2 yS(x) =0:<br />
From (16) <strong>and</strong> (17) <strong>on</strong>e obtains<br />
0=[S(x);T(x)]y[S(x);T(x)] + [S(x) 2 ;T(x)]yT(x)+[T (x);S(x)]T (x)yS(x) =<br />
[S(x);T(x)]y[S(x);T(x)] + ([S(x);T(x)]S(x)+S(x)[S(x);T(x)])yT(x)+<br />
which reduces to<br />
[T (x);S(x)]T (x)yS(x)<br />
(18) [S(x);T(x)]y[S(x);T(x)] + [T (x);S(x)]T (x)yS(x) =0:<br />
The substituti<strong>on</strong> yS(x)z for y in (18) gives<br />
(19) [S(x);T(x)]yS(x)z[S(x);T(x)] + [T (x);S(x)]T (x)yS(x)zS(x) =0:<br />
On the other h<strong>and</strong>, right multiplicati<strong>on</strong> of (18) by zS(x) leads to<br />
(20) [S(x);T(x)]y[S(x);T(x)]zS(x)+[T (x);S(x)]T (x)yS(x)zS(x) =0:<br />
From (19) <strong>and</strong> (20) we obtain<br />
(21) [S(x);T(x)]yA(x; z) =0;<br />
where A(x; z) st<strong>and</strong>s for [S(x);T(x)]zS(x) , S(x)z[S(x);T(x)]. The substituti<strong>on</strong><br />
zS(x)y for y in (21) gives<br />
(22) [S(x);T(x)]zS(x)yA(x; z) =0:<br />
Left multiplicati<strong>on</strong> of (21) by S(x)z leads to<br />
(23) S(x)z[S(x);T(x)]yA(x; z) =0:
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