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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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