Mathematics – Rings and Algebras
Scientific paper
2011-11-23
Mathematics
Rings and Algebras
7 pages
Scientific paper
Let D be a division ring. We say that D is left algebraic over a (not necessarily central) subfield K of D if every x in D satisfies a polynomial equation x^n + a_{n-1}x^{n-1}+...+a_0=0 with a_0,...,a_{n-1} in K. We show that if D is a division ring that is left algebraic over a subfield K of bounded degree d then D is at most d^2-dimensional over its center. This generalizes a result of Kaplansky. For the proof we give a new version of the combinatorial theorem of Shirshov that sufficiently long words over a finite alphabet contain either a q-decomposable subword or a high power of a non-trivial subword. We show that if the word does not contain high powers then the factors in the q-decomposition may be chosen to be of almost the same length. We conclude by giving a list of problems for algebras that are left algebraic over a commutative subring.
Bell Jason P.
Drensky Vesselin
Sharifi Yaghoub
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