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DOI: 10.46698/d4945-5026-4001-v
Полудифференцирования в первичных кольцах
Раза М. А. , Рехман Н.
Владикавказский математический журнал. 2021. Том 23. Выпуск 2.С.70-77.
Аннотация: Пусть \(\mathscr{R}\) - первичное кольцо с расширенным центроидом \(\mathscr{C}\) и с фактор-кольцо Матриндейла \(\mathscr{Q}\). Аддитивное отображение \(\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}\) называют полупроизводной, ассоциированной с \(\mathscr{G}: \mathscr{R}\rightarrow \mathscr{R}\), если \(\mathscr{F}(xy)=\mathscr{F}(x)\mathscr{G}(y)+x\mathscr{F}(y)= \mathscr{F}(x)y+\mathscr{G}(x)\mathscr{F}(y)\) и \(\mathscr{F}(\mathscr{G}(x))=\mathscr{G}(\mathscr{F}(x))\) для всех \(x,y\in \mathscr{R}\). В этой работе мы исследуем и описываем строение первичных колец \(\mathscr{R}\), удовлетворяющих условию \(\mathscr{F}(x^m\circ y^n)\in \mathscr{Z(R)}\) для всех \(x, y \in \mathscr{R}\), где \(m,n\in\mathbb{Z}^+\) и \(\mathscr{F}:\mathscr{R}\rightarrow\mathscr{R}\) - полупроизоводная с автоморфизмом \(\xi\) кольца \(\mathscr{R}\). Далее, в качестве приложения нашего теоретико-кольцевого результата мы обсуждаем природу \(\mathscr{C}^*\)-алгебр. Точнее, для любой примитивной \(\mathscr{C}^\ast\)-алгебры \(\mathscr{A}\). Точнее, для любой примитивной \(\mathscr{C}^\ast\)-алгебры \(\mathscr{A}\) получаем следующее. Если антиизоморфизм \(\zeta:\mathscr{A}\to\mathscr{A}\) удовлетворяет соотношению \((x^n)^\zeta+x^{n*}\in\mathscr{Z}(\mathscr{A})\) для всех \({x,y}\in \mathscr{A},\) то \(\mathscr{A}\) служит \(\mathscr{C}^{*}-\mathscr{W}_{4}\)-алгеброй, т.е., \(\mathscr{A}\) удовлетворяет стандартному тождеству \(\mathscr{W}_4(a_1,a_2,a_3,a_4)=0\) for all \(a_1,a_2,a_3,a_4\in\mathscr{A}\).
Образец цитирования: Raza M. A. and Rehman N. A Note on Semiderivations in Prime Rings and \(\mathscr{C}^*\)-Algebras // Владикавк. мат. журн. 2021. Т. 23, № 2. C. 70-77. DOI 10.46698/d4945-5026-4001-v
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