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Handbook of Practical Logic and Automated Reasoning by John Harrison

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The word problem for rings

To state the next result, it’s helpful to introduce the concept of an ideal in a polynomial ring.20 If p1, . . . , pn are polynomials in R[x1, . . . , xk] (we often abbreviate such a finite sequence of variables xi as image) we write IdR (p1, . . . , pn) (read ‘the ideal generated by p1, . . . , pn’) for the set of polynomials that can be expressed as follows:

image

where qi (sometimes referred to as cofactors) are arbitrary polynomials with coefficients in R, allowing the empty sum 0. With slight abuse of language, we will also ...

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