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Generalized polynomial and rational identities
Abstract
This dissertation includes several results in the theory of noncommutative rings. Namely: (1) We define an equivalence relation on the set of inner derivations of division rings of degree n for each positive integer n. We then show that for two equivalent derivations, $\delta\sb1$ and $\delta\sb2$ of $E\sb1$ and $E\sb2$ respectively, the sets of $\delta\sb1$-rational identities of $E\sb1$ and of $\delta\sb2$-rational identities of $E\sb2$ are the same. (2) For a division ring E, with center C and K an infinite subfield of C, we look at the generalized rational identities involving derivations, automorphisms, and antiautomorphisms of E where the mappings are all K-linear. We show that under certain conditions, the indeterminates with mappings can be replaced with new indeterminates, yielding a generalized rational identity without (or with a more restricted set of) mappings. (3) We prove: Let R be a prime ring of characteristic $\ne$ 2,3, with involution of the first kind. Let K denote the skew elements of R and C denote the extended centroid of R. Assume that (RC: C) $\ge$ 36. Then any Lie derivation of K into itself can be extended to a derivation of $\langle K\rangle,$ the associative subring generated by K.
Subject Area
Mathematics
Recommended Citation
Swain, Gordon A, "Generalized polynomial and rational identities" (1993). Doctoral Dissertations Available from Proquest. AAI9408352.
https://scholarworks.umass.edu/dissertations/AAI9408352