TY - JOUR AU - YEN, CHEN-TE PY - 1997/12/01 Y2 - 2024/03/28 TI - PRIME NONASSOCIATIVE RINGS WITH SKEW DERIVATIONS JF - Tamkang Journal of Mathematics JA - Tamkang J. Math. VL - 28 IS - 4 SE - Papers DO - 10.5556/j.tkjm.28.1997.4307 UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4307 SP - 309-312 AB - <div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPSMT'; vertical-align: 4.000000pt;">Let $R$ b</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPSMT';">e a prime nonassociative ring, $G$ the nucleus of $R$ and $s$, $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPSMT';">t$ b</span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT'; vertical-align: -3.000000pt;">e automorphisms </span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPSMT';">of $</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">R$.</span></p><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">(I) <span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">Suppose that $\delta$ is an </span></span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">$s$-derivation <span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT'; vertical-align: 3.000000pt;">of $R$ </span>such that $s\delta<span style="font-size: 14.000000pt; font-family: 'TimesNewRomanPSMT';">=\delta </span>s$ and $\lambda$ is an $t$-derivation of $<span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">R$</span>. </span><span style="font-size: 8.000000pt; font-family: 'ArialMT';">If $\lambda\delta^n=0$ and $\delta^n(R)\subset G$, </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT'; vertical-align: 3.000000pt;">where $n$ is a fixed </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">positive integer, then $\lambda=0$ or $\delta^{</span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">3n-1}=0$.</span></p><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">(II) </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">Assume that $\delta$ and $\lambda$ are derivations of $R$. </span><span style="font-size: 8.000000pt; font-family: 'ArialMT';">If </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">th</span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT'; vertical-align: -3.000000pt;">ere exists a fixed positive </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT'; vertical-align: -5.000000pt;">integer $n$ such </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">that $\lambda^n\delta=0$</span><span style="font-size: 8.000000pt; font-family: 'TimesNewRomanPSMT';">, </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">and $\delta(R)</span><span style="font-size: 9.000000pt; font-family: 'DFKaiShu-SB-Estd-BF';">\subset </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">G$ or $\lambda^n(R)\subset G$, then $\delta^2</span><span style="font-size: 9.000000pt; font-family: 'ArialMT';">= </span><span style="font-size: 8.000000pt; font-family: 'TimesNewRomanPSMT';">0$ </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">or $\lambda^{6n-4}</span><span style="font-size: 9.000000pt; font-family: 'ArialMT';">= 0$</span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">. </span></p></div></div></div> ER -