@article{YEN_1995, title={NONASSOCIATIVE RINGS WITH A SPECIAL DERIVATION}, volume={26}, url={https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4397}, DOI={10.5556/j.tkjm.26.1995.4397}, abstractNote={<div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">Let $</span><span style="font-size: 9.000000pt; font-family: ’Arial’; font-style: italic;">R$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">be a nonassociative ring, $</span><span style="font-size: 9.000000pt; font-family: ’Arial’; font-style: italic;">N$, $L$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">and $</span><span style="font-size: 9.000000pt; font-family: ’Arial’; font-style: italic;">G$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">the left nucleus, right nucleus and nucleus respectively. </span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPSMT’;">It </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">is shown that if $</span><span style="font-size: 9.000000pt; font-family: ’Arial’; font-style: italic;">R$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">is a prime ring with a derivafion $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">d$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">such that $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">ax+ d(x) \in</span> <span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">G$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">where $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">a \in</span> <span style="font-size: 9.000000pt; font-family: ’Arial’; font-weight: bold;">Z$, </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">the ring of rational integers, or $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">a \in</span> <span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">G$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">with $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">(ad)(x) </span><span style="font-size: 10.000000pt; font-family: ’ArialMT’;">= </span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">ad(x) </span><span style="font-size: 10.000000pt; font-family: ’ArialMT’;">= </span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">d(ax)$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">and </span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">$ax = xa$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">for all $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">x$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">in $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">R$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">then either $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">R$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">is associative or $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">ad+ </span><span style="font-size: 9.000000pt; font-family: ’Arial’; font-style: italic;">d^2</span> <span style="font-size: 11.000000pt; font-family: ’ArialMT’;">= </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">2d(R)^2</span> <span style="font-size: 11.000000pt; font-family: ’ArialMT’;">= </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">0$. This result is also valid under the weaker hypothesis </span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">$ax+ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">d(</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">x) \in</span> <span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">N </span><span style="font-size: 13.000000pt; font-family: ’TimesNewRomanPSMT’;">\cap </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">L$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">for all $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">x$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">in $</span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">R$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">for the simple ring case, and we obtain that either $</span><span style="font-size: 9.000000pt; font-family: ’Arial’; font-style: italic;">R$ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">is associative or $</span><span style="font-size: 10.000000pt; font-family: ’TimesNewRomanPS’; font-style: italic;">((ad+ </span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">d^2</span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">)(R))^2</span> <span style="font-size: 16.000000pt; font-family: ’TimesNewRomanPSMT’;">=</span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">0$ for the prime ring case. </span></p> </div> </div> </div>}, number={3}, journal={Tamkang Journal of Mathematics}, author={YEN, CHEN-TE}, year={1995}, month={Sep.}, pages={193–199} }