Topic

# Ideal (ring theory)

About: Ideal (ring theory) is a research topic. Over the lifetime, 6332 publications have been published within this topic receiving 72932 citations.

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30 May 2010

TL;DR: A fully homomorphic encryption scheme, using only elementary modular arithmetic, that reduces the security of the scheme to finding an approximate integer gcd, and investigates the hardness of this task, building on earlier work of Howgrave-Graham.

Abstract: We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct a fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of our scheme is the conceptual simplicity.
We reduce the security of our scheme to finding an approximate integer gcd – i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer. We investigate the hardness of this task, building on earlier work of Howgrave-Graham.

1,486 citations

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01 Jan 1990

TL;DR: In this paper, Dedekind Domains and Valuations have been used to define the theory of P-adic fields and to define a local compact Abelian group. But they do not consider the relation between the two types of fields.

Abstract: 1. Dedekind Domains and Valuations.- 2. Algebraic Numbers and Integers.- 3. Units and Ideal Classes.- 4. Extensions.- 5. P-adic Fields.- 6. Applications of the Theory of P-adic Fields.- 7. Analytical Methods.- 8. Abelian Fields.- 9. Factorizations 9.1. 485Elementary Approach.- Appendix I. Locally Compact Abelian Groups.- Appendix II. Function Theory.- Appendix III. Baker's Method.- Problems.- References.- Author Index.- List of Symbols.

984 citations

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TL;DR: The invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G-space X were studied in this paper.

Abstract: Let G be a compact Lie group (e.g., a finite group) and let HG= H*(BG, Z/pZ) be its mod p cohomology ring. One knows this ring is finitely generated, hence upon dividing out by the ideal of nilpotent elements it becomes a finitely generated commutative algebra over the field Z/pZ. It is the purpose of this series of papers to relate the invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G. For example we prove a conjecture of Atiyah and Swan to the effect that the Krull dimension of the ring equals the maximum rank of an elementary abelian p-subgroup. Another result, which will appear in part II, asserts that the minimal prime ideals of the ring are in one-one correspondence with the conjugacy classes of maximal elementary abelian p-subgroups. Actually the theorems of the series are formulated more generally for the equivariant cohomology ring of a G-space X, defined by the formula

719 citations

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TL;DR: In this article it was shown that a projective module P has the finite exchange property if and only if, whenever P = N + M where N and M are submodules, there is a decomposition P = A @ B with A S N and B C M.

Abstract: Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x e R with x-x2 cL, there exists an idempotent e c R such that e x E L. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever P = N + M where N and M are submodules, there is a decomposition P = A @ B with A S N and B C M. In 1972 Warfield showed that if M is a module over an associative ring R then M has the finite exchange property if and only if end M has the exchange property as a module over itself. He called these latter rings exchange rings and showed (using a deep theorem of Crawley and Jonsson) that every projective module over an exchange ring is a direct sum of cyclic submodules. Let J(R) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo J(R), then R is an exchange ring and so generalized theorems of Kaplansky and Muller. The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring if and only if idempotents can be lifted modulo every left (respectively right) ideal. The properties of these rings are examined in the first section and the theorem is proved in the second section. The theorems of Warfield are then easily deduced and a new condition that a projective module have the finite exchange property is given. 1. Suitable rings. In this section, the rings of interest are defined, some of their properties are deduced, and several examples are given. All rings are assumed to be associative with identity and J(R) denotes the Jacobson radical of a ring R. 1.1. PROPOSITION. If R is a ring, the following conditions are equivalent for an element x of R. Received by the editors December 2, 1975. AMS (MOS) subject classifications (1970). Primary 16A32, 16A64; Secondary 16A30, 16A50.

662 citations

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TL;DR: In this paper, it was shown that if I can be generated by r elements, then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r.

Abstract: 0. Introduction. Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. A classical result of Krull is that if I can be generated by r elements then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r. If, moreover, the grade of I (the length of the longest R-sequence contained in I) is r, then I enjoys certain special properties summarized in the term "perfect" as used by iRees [30, p. 32]: I is perfect if the homological (or projective) dimension of R/I as an R-module is equal to the grade of I. The associated primes of a perfect ideal I all have the same grade as I, that is, perfect ideals are grade unmixed. If R is Cohen-Macaulay, the grade of any ideal is equal to its little rank of height (the least rank of any minimal prime) ; in particular, the notions of grade and rank coincide on -primes, and perfect ideals are rank unmixed. Moreover, if I is perfect in a Cohen-Macaulay ring R, R/I is again (Cohen-Macaulay. Macaulay's famous theorem that in a polynomial ring over a field a rank r ideal which can be generated by r elements is rank unmixed [36, p. 203] is then a consequence of two facts: a polynomial ring over a field is CohenMacaulay, and a grade r ideal generated by r elements is perfect. This is the classical example of a perfect ideal. Good discussions of the subject. are available: see [9], [24, ? 25], [30], [18, Ch. 3], and [36, Appendix 6]. The Noetherian restriction on R is, for certain purposes, unnecessary in the discussion of perfect ideals, if one adopts a suitable definition of grade. This idea is worked out in [1]. Suppose that R is (locally) regular, and I is an ideal of R such that R/I is not the direct product of two rings in a nontrivial way. Then I is perfect if and only if R/I is Cohen-Macaulay. In particular, this is the situation when R is a polynomial ring over a field and I is homogeneous. It is very natural, then, to hunt for perfect ideals. Relatively few classes are known, but several authors [4, 6, 8, 33] have established the perfection

439 citations