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2. A prime element of a ring is an element that generates a prime ideal.
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A ring is prime if and only if its zero ideal is a prime ideal.
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He proved that every cyclotomic integer can be factorised uniquely into prime ideal numbers.
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Every prime ideal of A is principal.
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In particular, there is a notion of a ‘prime ideal’, a reasonable and tractable analogue of a prime number.
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Kummer also proved Fermat's last theorem for a considerable class of prime exponents (see regular prime, ideal class group).
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Prove that the factor ring R/J is an integral domain if and only if J is a prime ideal.
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However, the proofs of both statements require the Boolean prime ideal theorem, a weak form of the axiom of choice.
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In the ring Z of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal.
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More precisely, given a prime ideal of a base field k, this ideal does not remain prime in an extension K of k.
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The quotient ring is the product OK/pOK = Fp × Fp. p is ramified (p) is the square of a prime ideal of OK.
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It also affects the study of infinite groups to a large extent, as well as ring and order theory (see Boolean prime ideal theorem).
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In commutative ring theory, numbers are often replaced by ideals, and the definition of the prime ideal tries to capture the essence of prime numbers.
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More precisely, the spectrum of a commutative ring is the space of its prime ideals equipped with Zariski topology, and augmented with a sheaf of rings.
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The points, curves, surfaces and so on from classical geometry are replaced in the non-commutative world by a spectrum of prime ideals and representation theory.
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That this approach is more than a mere curiosity was demonstrated by characterising generating sets for prime ideals of a special but important type, called H-prime ideals.
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Formally, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets.
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More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization.
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The wide, dried mudflats of Antiga Prime were ideal for speed and lousy for cover.
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In the years preceding the start of the RTQASL project, it was found that quantum matrices, as well as their quotients by certain prime ideals, could be constructed using a weighted grid-like network with weights in non-commutative algebras.
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Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC).
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The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.
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Intelligence reports that site five is the ideal prime target.
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Prime factorisation of ideals is much tidier: both existence and uniqueness are valid.
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He conformed to the prime minister's beau ideal of the gentleman warrior.
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700 sinne gevind in 20 ms. Hulle kom uit baie bronne en word nie nagegaan nie.