By Ash R.B.

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**Sample text**

The kernel of g is P RP , so by the ﬁrst isomorphism theorem, g(RP ) ∼ = RP /P RP , a ﬁeld (because P RP is a maximal ideal). Thus we may replace (R, h) by (RP , g). Our ﬁrst step is to extend h to a homomorphism of polynomial rings. If f ∈ R[x] with h(ai )xi ∈ F [x]. Let I = {f ∈ R[x] : f (α) = 0}. Then f (x) = ai xi , we take h(f ) = J = h(I) is an ideal of F [x], necessarily principal. Say J = (j(x)). If j is nonconstant, it must have a root β in the algebraically closed ﬁeld C. We can then extend h to h : R[α] → C via h(α) = β, as desired.

This relationship suggests that we look for various ways of deﬁning the dimension of an arbitrary commutative ring. We will see that under appropriate hypotheses, several concepts of dimension are equivalent. Later, we will connect the algebraic and geometric ideas. 1 The Calculus of Finite Diﬀerences Regrettably, this charming subject is rarely taught these days, except in actuarial programs. It turns out to be needed in studying Hilbert and Hilbert-Samuel polynomials in the next section. 1 Lemma Let g and G be real-valued functions on the nonnegative integers, and assume that ∆G = g, that is, G(k + 1) − G(k) = g(k) for all k ≥ 0.

Explicitly, g(x) = (gi (x)), and the coherence of the gi forces the sequence (gi (x)) to be coherent. An inverse limit of an inverse system of rings can be constructed in a similar fashion, as coherent sequences can be multiplied componentwise, that is, (xi )(yi ) = (xi yi ). 2 Examples 1. Take R = Z, and let I be the ideal (p) where p is a ﬁxed prime. Take Mn = Z/I n and θn+1 (a + I n+1 ) = a + I n . The inverse limit of the Mn is the ring Zp of p-adic integers. 2. COMPLETION OF A MODULE 5 2. Let R = A[x1 , .

### A course in commutative algebra by Ash R.B.

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