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# Maximal ideal

About: Maximal ideal is a research topic. Over the lifetime, 2449 publications have been published within this topic receiving 31683 citations.

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TL;DR: In this article, Grothendieck showed that a covariant functor F from C to Sets is pro-representable if it has the form 1.0, 2.0.

Abstract: 0. Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation. Let A be a complete noetherian local ring, ,u its maximal ideal, and k = A/l the residue field. (In most applications A is k itself, or a ring of Witt vectors.) Let C be the category of Artin local A-algebras with residue field k. A covariant functor F from C to Sets is called pro-representable if it has the form

780 citations

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TL;DR: In this paper, the authors studied the structure and properties of injective modules, particularly over Noetherian rings, and showed that if a module M has a maximal, injective submodule C (as is the case for left-Noetherian ring), then C contains a carbon-copy of every injective module of M, and MjC has no submodules different from 0.

Abstract: Introduction In this discussion every module over a ring R will be understood to be a left i2-module. R will always have a unit, and every module will be unitary. The aim of this paper is to study the structure and properties of injective modules, particularly over Noetherian rings. B. Eckmann and A. Schopf have shown that if M is a module over any ring, then there exists a unique, minimal, injective module E(M) containing it. The module E(M) will be a major tool in our investigations, and we shall systematically exploit its properties. In § 1 we show that if a module M has a maximal, injective submodule C (as is the case for left-Noetherian rings), then C contains a carbon-copy of every injective submodule of M, and MjC has no injective submodules different from 0. Although C is unique up to an automorphism of My C does not in general contain every injective submodule of M, In fact, the sum of two injective submodules of a module is always injective if and only if the ring is left-heredita ry. In § 2 we show that for any ring R a module E is an indecomposable, injective module if and only if E = E(R\J)y where J is an irreducible, left ideal of R. We prove that if R is a left-Noetherian ring, then every injective Jϋ-module has a decomposition as a direct sum of indecomposable, injective submodules. Strong uniqueness assertions can be made concerning such decompositions over any ring. In § 3 we take R to be a commutative, Noetherian ring, and P to be a prime ideal of R. We prove there is a one-to-one correspondence between the prime ideals of R and the indecomposable, injective Rmodules given by P**E(RjP). We examine the structure of the module E = E{RjP)y and show that if At is the annihilator in E of P\ then E = U At and -4ί+1/At is a finite dimensional vector space over the quotient field of R/P. The ring of iϋ-endomorphi sms of E is isomorphic in a natural way to Rp, the completion of the ring of quotients of R with respect to R-P. As an ^-module E is an injective envelope of RpjP, where P is the maximal ideal of Rp. If P is a maximal ideal of Ry then E is a countably generated β-module. Every indecomposable, injective i2-module is finitely generated if and only if R has the minimum condition on ideals. In § 4 we take R to be a commutative, Noetherian, complete, local ring, P the maximal ideal of R and E = E{RjP). Then the eontravariant,

628 citations

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01 Apr 1954TL;DR: In the analytic theory of ideals, a reduction is defined as follows: if and are ideals and ⊆, then is called a reduction of if n = n + 1 for all large values of n. The usefulness of the concept depends mainly on two facts as mentioned in this paper.

Abstract: This paper contains some contributions to the analytic theory of ideals. The central concept is that of a reduction which is defined as follows: if and are ideals and ⊆ , then is called a reduction of if n = n +1 for all large values of n . The usefulness of the concept depends mainly on two facts. First, it defines a relationship between two ideals which is preserved under homomorphisms and ring extensions; secondly, what we may term the reduction process gets rid of superfluous elements of an ideal without disturbing the algebraic multiplicities associated with it. For example, the process when applied to a primary ideal belonging to the maximal ideal of a local ring gives rise to a system of parameters having the same multiplicity; but the methods work almost equally well for an arbitrary ideal and bring to light some interesting facts which are rather obscured in the special case. The concept seems to be suitable for a variety of applications. The present paper contains one instance which is a generalized form of the associative law for multiplicities (see § 8), and the authors hope to give other illustrations in a separate paper.

624 citations

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TL;DR: In this paper, the authors define a graded ring K, F associated to any rational function field F and construct a homomorphism associated with a discrete valuation on F with residue class field F.

Abstract: The first section of this paper defines and studies a graded ring K . F associated to any field F. By definition, K~F is the target group of the universal n-linear function from F ~ x ... • F ~ to an additive group, satisfying the condition that al • " ' x a, should map to zero whenever a i -q-a i + ~ = 1. Here F ~ denotes the multiplicative group F 0 . Section 2 constructs a homomorphism ~: K,F---, K~__I_~ associated with a discrete valuation on F with residue class field F. These homomorphisms ~ are used to compute the ring K, F(t) of a rational function field, using a technique due to John Tate. Section 3 relates K . F to the theory of quadratic modules by defining certain " Stiefel-Whitney invariants" of a quadratic module over a field F of characteristic . 2 . The definition is closely related to Delzant [-5]. Let W be the Witt ring of anisotropic quadratic modules over F, and let I c W be the maximal ideal, consisting of modules of even rank. Section 4 studies the conjecture that the associated graded ring

539 citations

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TL;DR: In this paper, it was shown that every TCzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on E is continuous in either the weak, ultra-weak, strong, or ultrastrong topologies if and only if it is complete in all four simultaneously.

Abstract: Let 5C be a complex Hubert Space, J3(3C) the ring of bounded operators on 3C, E an abelian symmetric subring of B(3Z) containing the identity which is closed in the weak operator topology, E\ the commutant of E, and suppose E\ has a cyclic vector £o which we normalize so that |£o| = 1 . Diximier [ l] has shown that E (respect. Ei), as a Banach space, is the dual of the Banach space R (respect. Ri) of all linear forms on E (respect. E\) that are continuous in the ultra-strong topology of E (respect. Ei). In this note we show that every TCzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on E is continuous in either the weak, ultraweak, strong, or ultrastrong topologies if and only if it is continuous in all four simultaneously. In the process, we obtain an integral representation for such T, which we later use in a theorem on centrally reducible positive functionals on E%. We denote the maximal ideal space of E by M, and for A, B, • • • £ E , we denote the corresponding Gel'fand transforms by a, 6, • • • . Then A—>a is an isometric isomorphism from £ onto C(M). Consequently, every bounded linear functional on E has the form

462 citations