Semiprime rings with differential identities. (English) Zbl 0769.16017

Let \(R\) be a semi-prime ring with maximal right quotient ring \(U\) and let \(\text{Der}(U)\) be the set of derivations of \(U\). The extended centroid of \(R\) is \(C\), the center of \(U\). A differential polynomial is an element \(f \in U*_ C C\{X^ W\}\), the free product over \(C\) of \(U\) and the free \(C\)-algebra in indeterminates \(x_ i^ w\), where each exponent corresponds to a word \(d_ 1\dots d_ n\) with \(d_ j \in \text{Der}(U)\). Such an \(f(x_ i)\) is a differential identity on a subset \(T\) of \(U\) if \(f(t_ i) = 0\) for all substitutions from \(T\).
The main result of the paper is that if \(I_ R\) is a dense \(R\)-submodule of \(U_ R\), then \(I\) and \(U\) have the same differential identities. One consequence of this is that if \(f(x_ i)\) is a differential identity on an ideal \(I\) of \(R\), then \(f(u_ i)I = 0\) for all \(u \in U\). The main result is an extension to maximal quotient rings of a theorem of V. K. Kharchenko [Algebra Logika 18, 86-119 (1979; Zbl 0464.16027)], and applies this result when \(R\) is a prime ring. The extension to semi-prime rings uses results of K. I. Bejdar [Vestn. Mosk. Univ., Ser. I 1978, No. 5, 36-43 (1978; Zbl 0403.16003)] on orthogonal completeness.


16W25 Derivations, actions of Lie algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16N60 Prime and semiprime associative rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)