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By M. Scheunert

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It is clear that (a)⇒(b)⇒(c). When the assertion (c) is true, the exact sequence 0 → Ker(f ) → M → N → 0 splits and M is isomorphic to N ⊕ Ker(f ); since M and N everywhere have the same rank, we conclude that Ker(f ) = 0. 40 Chapter 1. 6) Example. Let F be a field and K = F [[t]] the ring of formal series; it is a local ring and its maximal ideal m is the ideal generated by the indeterminate t; the residue field K/m can be identified with F . The multiplication by t is an injective mapping K → K; but after the extension K → F it gives the null morphism F → F .

4) show several examples of local properties. 5) Corollary. For a K-module M the following assertions are equivalent: (a) M = 0; (b) Mp = 0 for every prime ideal p; (c) Mm = 0 for every maximal ideal m. 6) Corollary. When N and N are submodules of M , the following assertions are equivalent: (a) N ⊂ N ; (b) Np ⊂ Np for every prime ideal p; (c) Nm ⊂ Nm for every maximal ideal m. We get another triplet of equivalent assertions if we replace the inclusions with equalities. Proof. It is clear that (a)⇒(b)⇒(c).

Algebraic Preliminaries All these statements are evident except perhaps the last one, a consequence of the inclusions V(a) ∪ V(b) ⊂ V(a ∩ b) ⊂ V(ab) ⊂ V(a) ∪ V(b). 1) proves that there is a topology on Spec(K) for which the closed subsets are the subsets V(a); it is called the Zariski topology of Spec(K). For every p ∈ Spec(A) the topological closure of {p} is V(p); thus the point {p} is closed if and only if p is a maximal ideal; this topology is almost never a Hausdorff topology. 10, with each p ∈ Spec(K) is associated a localized ring Kp with maximal ideal pKp , and a residue field Kp /pKp.

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