By George Boole

ISBN-10: 140218364X

ISBN-13: 9781402183645

This Elibron Classics ebook is a facsimile reprint of a 1877 version by means of Macmillan and Co., London.

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**Extra resources for A treatise on differential equations**

**Example text**

We say that A is ﬁnite over A0 if it is a ﬁnitely generated A0 -module. The composition of two ﬁnite homomorphisms is clearly a ﬁnite homomorphism. It is easy to see that A is ﬁnite over A0 if and only if it is integral and ﬁnitely generated over A0 . 9 (Noether normalization lemma). Let A be a non-zero ﬁnitely generated algebra over a ﬁeld k. Then there exist an integer d ≥ 0 and a ﬁnite injective homomorphism k[T1 , . . , Td ] → A. Proof By assumption, we have A = k[X1 , . . , Xn ]/I. We will use induction on n.

Tn ]/I be a ﬁnitely generated algebra over k. Then there is a bijection between the closed points of Spec A and the algebraic set Z(I) := {(α1 , . . , αn ) ∈ k n P (α1 , . . , αn ) = 0, for every P (T ) ∈ I}. Proof We can identify the closed points of Spec A with the maximal ideals of k[T1 , . . , Tn ] containing I. Let m = (T1 − α1 , . . , Tn − αn ) be a maximal ideal of k[T1 , . . , Tn ]. Let P (T ) ∈ k[T1 , . . , Tn ]. Using the Taylor expansion of P (T ) at α := (α1 , . . , αn ), we see that P (T ) ∈ m if and only if P (α) = 0.

Proof In fact, the ﬁltration (Mn ∩ N )n induces on N the structure of a submodule of M . 10. 12 (The Artin–Rees lemma). Let A be a Noetherian ring, I an ideal of A, M a ﬁnitely generated A-module, and N a submodule of M . Then there exists an n0 such that (I n+1 M ) ∩ N = I((I n M ) ∩ N ) for every n ≥ n0 . 13. Let A be a Noetherian ring, I an ideal of A, and M a ﬁnitely generated A-module. Then ∩n≥0 (I n M ) is the set of elements x ∈ M for which there exists an α ∈ I such that (1 + α)x = 0. Proof If (1 + α)x = 0 for an α ∈ I, then x = −xα ∈ IM .

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