Prime radicals of constants of algebraic derivations.
Let \(R\) be a prime algebra over a field \(F\), \(Q\) the symmetric quotient ring of \(R\), and \(C\) the extended centroid of \(R\). Any derivation of \(R\) extends to \(Q\), and any derivation \(\delta\) of \(Q\) is called a continuous derivation of \(R\) if \(\delta(I)\subseteq R\) for a nonzero ideal \(I\) of \(R\); \(\delta\) is an \(F\)-derivation if \(\delta(F)=0\). If \(\delta\) is an \(F\)-continuous derivation of \(R\) that is algebraic over \(F\), then it is well known that there is \(b\in Q\) so that \(\delta=ad(b)\) or \(\text{char\,}R=p\geq 2\) and \(g(\delta)=ad(b)\) for \(g(x)=h(x^p)\in F[x]\) with \(g(0)=0\); further, \(b\) is algebraic over \(F\) with minimal polynomial \(m(x)\in F[x]\). Finally, set \(R^{(\delta)}=\{x\in R\mid\delta(x)=0\}\) and for any ring \(S\) let \(P(S)\) be its prime radical. The main results in the paper relate the degree of nilpotence of \(P(R^{(\delta)})\) to the powers of the factors of \(m(x)\) in \(F[x]\). Specifically, if \(t\) is the largest power of any irreducible factor appearing in \(m(x)\) then \(P(R^{(\delta)})^{{2^t}-1}=0\), and for any overring \(S\) of \(P(R^{(\delta)})\) contained in \(C(b)=\{q\in Q\mid bq=qb\}\), \(P(R^{(\delta)})=R^{(\delta)}\cap P(S)\). The result as stated by the authors describes a potentially smaller exponent than \(t\). They also show that \(P(C(b))^{{2^t}-1}=0\).
scientific article; zbMATH DE number 5702943