By Francis Borceux

Lecture notes in arithmetic No.1348

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Example text

When R is an arbitrary commutative ring the set {f/(F~0)}i/>0 is not, in general, a complete set of invariants for the similarity class of 99. 14] can be found a brief exposition, based on [94,53,54], on similarity of matrices when R is a local ring or R is the ring of integers. 2. 1. Trace and determinant The natural bilinear mapping Let M be an R-module and M* the dual module of M. ,n) where qgi,,m)(m') = u(m')m, induces a canonical homomorphism OM : M* @R M ~ EndR(M). PROPOSITION 54. Let M be a finitely generated projective R-module.

This fact and (I) =r (II) are proved in [ 1]. When Min(R) with the Zariski topology is quasi-compact then (I) r162(II). Other characterizations for the integrally-closeness of R[T] are given in [85,86]. 2. Priifer rings By Proposition 29 the existence of a solution for a system of linear equations is a local property. Using this fact one can prove, see [67], the following result which is an extension of Theorem 34 to arbitrary systems. THEOREM 37. Let R be a commutative ring. e. every finitely generated ideal o f R is flat).

Xi , for i -- I, 2 . . . ~'. ,+... +,,, + I fori--l,2 -- u i , . . . j 6 {I, 2 . . . n} \ {gl + K2 + . . , - I . E3 COROLLARY I 12. Let r be a B r u n o v s k y linear system. Then Z is coefficient a s s i g n a b l e a n d h e n c e p o l e assignable. As a c o n s e q u e n c e of this result one has that if R is a field then a r e a c h a b l e s y s t e m is coefficient assignable. T h e f o l l o w i n g result, called the p o l e - s h i f t i n g t h e o r e m , c h a r a c t e r i z e s w h a t p o l y n o m i a l s are assignable to a non r e a c h a b l e s y s t e m o v e r a field.