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A formal inverse to the Cayley-Hamilton theorem. (English) Zbl 0618.16014
A Q-algebra with trace is defined to be an algebra R together with an R- linear map tr: $$R\to R$$, such that, for a,b$$\in R:$$ (i) $$tr(a)b=btr(a)$$, (ii) $$tr(ab)=tr(ba)$$, (iii) $$tr(tr(a)b)=tr(a)tr(b)$$. For such an algebra R, n a positive integer, R is said to satisfy the nth Cayley-Hamilton identity if, for all $$r\in R$$, $$\chi_ r^{(n)}(r)=0$$ where $\chi_ r^{(n)}(t)=\prod^{n}_{i=1}(t-t_ i)\quad and\quad \sum^{n}_{i=1}t^ k_ i=tr(r^ k).$ The main theorem of this paper is: If R is a Q-algebra with trace satisfying the nth Cayley Hamilton identity, then R can be embedded as a subring of the ring $$M_ n(A)$$ over a commutative ring A.
Reviewer: M.Beattie

MSC:
 16Rxx Rings with polynomial identity 16S50 Endomorphism rings; matrix rings
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References:
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