Supercentralizing automorphisms on prime superalgebras.
The author proves a superalgebra version of [\textit{J. H. Mayne}, Can. Math. Bull. 27, 122-126 (1984; Zbl 0537.16029)]. Here \(A\) is a superalgebra over a commutative ring \(K\) with \(1/2\in K\). The super-commutator, \([a,b]_s\), of \(a,b\in A\) is the extension of \([a,b]_s=ab-ba\) when \(a\) or \(b\) is an even element, and \([a,b]_s=ab+ba\) when both are odd. Let \(Z_s(A)=\{a\in A\mid [a,b]_s=0\) for all \(b\in A\}\), and \(\varphi\in\Aut(A)\) that preserves the grading. The first main result assumes that \(A\) is noncommutative and prime (the product of nonzero graded ideals is not zero) and that \([\varphi(a),a]_s\in Z_s(A)\) for all \(a\in A\) and proves that \(\varphi=I_A\) (the identity map), and if \(A\) has elements of odd degree then \(A\) is an order in a quaternion algebra. Assuming instead that \([\varphi(a),a]_s=0\) for all \(a\in A\) forces \(A\) to have no odd degree elements, and then if \(A\) is not commutative, again \(\varphi=I_A\).
scientific article; zbMATH DE number 5640595