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  2. Localization (commutative algebra) - Wikipedia

    en.wikipedia.org/wiki/Localization_(commutative...

    The localization of a commutative ring R by a multiplicatively closed set S is a new ring whose elements are fractions with numerators in R and denominators in S.. If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers.

  3. Integrally closed domain - Wikipedia

    en.wikipedia.org/wiki/Integrally_closed_domain

    Such a ring is necessarily a reduced ring, [5] and this is sometimes included in the definition. In general, if A is a Noetherian ring whose localizations at maximal ideals are all domains, then A is a finite product of domains. [6] In particular if A is a Noetherian, normal ring, then the domains in the product are integrally closed domains. [7]

  4. Chirplet transform - Wikipedia

    en.wikipedia.org/wiki/Chirplet_transform

    In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. [ 2 ] [ 3 ] Similar to the wavelet transform , chirplets are usually generated from (or can be expressed as being from) a single mother chirplet (analogous to the so-called mother wavelet of wavelet theory).

  5. Cohen–Macaulay ring - Wikipedia

    en.wikipedia.org/wiki/Cohen–Macaulay_ring

    This ring can also be described as the coordinate ring of the cuspidal cubic curve y 2 = x 3 over K. The subring K[t 3, t 4, t 5] of the polynomial ring K[t], or its localization or completion at t=0, is a 1-dimensional domain which is Cohen–Macaulay but not Gorenstein. Rational singularities over a field of characteristic zero are Cohen ...

  6. Local ring - Wikipedia

    en.wikipedia.org/wiki/Local_ring

    If K were indeed the function field of an algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or more there is a difficulty that is seen this way: if F and G are rational functions on V with F(P) = G(P) = 0, the function F/G. is an indeterminate ...

  7. Polynomial ring - Wikipedia

    en.wikipedia.org/wiki/Polynomial_ring

    Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.

  8. Nakayama's lemma - Wikipedia

    en.wikipedia.org/wiki/Nakayama's_lemma

    Let R be a ring that is graded by the ordered semigroup of non-negative integers, and let + denote the ideal generated by positively graded elements. Then if M is a graded module over R for which M i = 0 {\displaystyle M_{i}=0} for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative ...

  9. Laurent polynomial - Wikipedia

    en.wikipedia.org/wiki/Laurent_polynomial

    The ring of Laurent polynomials is a subring of the rational functions. The ring of Laurent polynomials over a field is Noetherian (but not Artinian ). If R {\displaystyle R} is an integral domain , the units of the Laurent polynomial ring R [ X , X − 1 ] {\displaystyle R\left[X,X^{-1}\right]} have the form u X k {\displaystyle uX^{k ...