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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.

##### MSC:
 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)