Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Canonical Basis for Ideals in a Polynomial Domain Over a Commutative Ring with Finite Basis for Ideals


     

   Subscribe/Renew Journal


Let A be a commutative ring with unit element and A[x] denote the ring of polynomials in x with coefficients from A. Hilbert's theorem is : -

If every ideal in A has a finite basis, then every ideal in A[x] also has a finite basis.

The proof of this as given by Van der Waerden in his Moderne Algebra Bd. II gives us no information beyond the fact that every ideal in A [x] has a finite basis. If we just reverse the argument in his proof we will be able to give actually a canonical basis for every ideal in A[x] which will be found to be a powerful tool in several applications.


Subscription Login to verify subscription
User
Notifications
Font Size


Abstract Views: 201

PDF Views: 0




  • Canonical Basis for Ideals in a Polynomial Domain Over a Commutative Ring with Finite Basis for Ideals

Abstract Views: 201  |  PDF Views: 0

Authors

Abstract


Let A be a commutative ring with unit element and A[x] denote the ring of polynomials in x with coefficients from A. Hilbert's theorem is : -

If every ideal in A has a finite basis, then every ideal in A[x] also has a finite basis.

The proof of this as given by Van der Waerden in his Moderne Algebra Bd. II gives us no information beyond the fact that every ideal in A [x] has a finite basis. If we just reverse the argument in his proof we will be able to give actually a canonical basis for every ideal in A[x] which will be found to be a powerful tool in several applications.