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Algebraic Geometry

By Antoine Chambert-Loir

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Then M is artinian if and only if both P and M/P are artinian. In particular, finite direct sums of artinian modules are artinian. 3. — Let A be a ring. An A-module M is said to be simple if its only submodules are {0} and M; this is equivalent to the existence of a maximal ideal m of A such that M ≃ A/m. The length of an A-module M is the dimension of the partially ordered set of its submodules. It is denoted by lengthA (M), or even length(M) if the ring A is clear from the context. 4). — Let M be an A-module and let N be a submodule of M.

Let A be a ring. The following properties are equivalent: (i) The ring A is artinian; (ii) The A-module A has finite length; (iii) The ring A is noetherian and dim(A) = 0. Proof. — Condition (ii) implies that every sequence of ideals of A which is either strictly increasing or strictly decreasing is finite, hence that A is artinian (condition (i)) and noetherian (the first half of condition (iii)). 6 that every prime ideal of A is maximal, hence dim(A) = 0. Let us assume that A is noetherian and that dim(A) = 0.

Surjective, resp. bijective. A similar definition applies for contravariant functors. A functor F is essentially surjective if for every object P of D, there exists an object M of C such that F(M) is isomorphic to P in the category D. 3) (Forgetful functors). — Many algebraic structures are defined by enriching other structures. Often, forgetting this enrichment gives rise to a functor, called a forgetful functor. For example, a group is already a set, and a morphism of groups is a map. There is thus a functor that associates to every group its underlying set, thus forgetting the group structure.

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