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searching for Quotient ring 26 found (157 total)

Artinian ideal (387 words) [view diff] exact match in snippet view article find links to article

Artinian ideal is an ideal I in R for which the Krull dimension of the quotient ring R/I is 0. Also, less precisely, one can think of an Artinian ideal as

Quadratic field (1,288 words) [view diff] exact match in snippet view article find links to article

{\displaystyle p} is inert ( p ) {\displaystyle (p)} is a prime ideal. The quotient ring is the finite field with p 2 {\displaystyle p^{2}} elements: O K / p

Jacobson ring (829 words) [view diff] exact match in snippet view article find links to article

Goldman ideal is maximal. Every quotient ring of R by a prime ideal has a zero Jacobson radical. In every quotient ring, the nilradical is equal to the

Function field (scheme theory) (597 words) [view diff] exact match in snippet view article

by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce

Perfect ideal (243 words) [view diff] exact match in snippet view article find links to article

such that its grade equals the projective dimension of the associated quotient ring. grade(I)=projdim⁡(R/I).{\displaystyle {\textrm {grade}}(I)={\textrm

Imaginary unit (4,087 words) [view diff] exact match in snippet view article find links to article

and so there is a quotient ring R [ x ] / ⟨ x 2 + 1 ⟩ . {\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .} This quotient ring is isomorphic to the

Levitzky's theorem (465 words) [view diff] exact match in snippet view article find links to article

Noetherian, a maximal nilpotent ideal N exists. By maximality of N, the quotient ring R/N has no nonzero nilpotent ideals, so R/N is a semiprime ring. As

Principal indecomposable module (501 words) [view diff] exact match in snippet view article find links to article

modular characters of simple modules represent both a subring and a quotient ring. The representation ring over the complex field is usually better understood

Regular sequence (1,229 words) [view diff] exact match in snippet view article find links to article

multiple of p. The element p is a non-zero-divisor in Z(p), and the quotient ring of Z(p) by the ideal generated by p is the field Z/(p). Therefore p

Torsion-free module (591 words) [view diff] exact match in snippet view article find links to article

condition is the zero module. Over a commutative ring R with total quotient ring K, a module M is torsion-free if and only if Tor1(K/R,M) vanishes. Therefore

Zero divisor (1,756 words) [view diff] exact match in snippet view article find links to article

in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set

Double centralizer theorem (920 words) [view diff] exact match in snippet view article find links to article

Therefore, by an isomorphism theorem for rings, RM is isomorphic to the quotient ring R/Ann(MR). Clearly when M is a faithful module, R and RM are isomorphic

Abstract analytic number theory (1,199 words) [view diff] exact match in snippet view article find links to article

element OK and the norm of an ideal I is given by the cardinality of the quotient ring OK/I. In this case, the appropriate generalisation of the prime number

Principal ideal ring (1,282 words) [view diff] exact match in snippet view article find links to article

is, if I is a proper ideal of principal right ideal ring R, then the quotient ring R/I is also principal right ideal ring. This follows readily from the

AKS primality test (2,448 words) [view diff] exact match in snippet view article find links to article

] {\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]} . Evaluating in a quotient ring of ( Z / n Z ) [ X ] {\displaystyle (\mathbb {Z} /n\mathbb {Z} )[X]}

Gorenstein ring (1,662 words) [view diff] exact match in snippet view article find links to article

regular sequence a1,...,an in the maximal ideal of R such that the quotient ring R/( a1,...,an) is Gorenstein of dimension zero. For example, if R is

Construction of the real numbers (4,049 words) [view diff] exact match in snippet view article find links to article

Then B has a unique maximal ideal I, the infinitesimal numbers. The quotient ring B/I gives the field R of real numbers [citation needed]. Note that B

Ring learning with errors signature (2,546 words) [view diff] exact match in snippet view article find links to article

2017 update to the GLP scheme called GLYPH. A RLWE-SIG works in the quotient ring of polynomials modulo a degree n polynomial Φ(x) with coefficients in

Cyclic redundancy check (5,953 words) [view diff] exact match in snippet view article find links to article

polynomial will result in a certain proportion of missed errors, due to the quotient ring having zero divisors. The advantage of choosing a primitive polynomial

Higher local field (1,382 words) [view diff] exact match in snippet view article find links to article

ring and a full flag of prime ideals such that their corresponding quotient ring is regular. A series of completions and localisations take place as

Complex number (11,600 words) [view diff] exact match in snippet view article find links to article

{R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C} } between the quotient ring and C {\displaystyle \mathbb {C} } . Some authors take this as the definition

HEAAN (1,734 words) [view diff] exact match in snippet view article find links to article

{\displaystyle q} , let R q := R / q R {\displaystyle R_{q}:=R/qR} be the quotient ring of R {\displaystyle R} modulo q {\displaystyle q} . Let χ s {\displaystyle

Restricted Lie algebra (2,328 words) [view diff] exact match in snippet view article find links to article

X\in {\mathfrak {g}}} ; then the restricted enveloping algebra is the quotient ring u ( g ) = U ( g ) / I {\displaystyle u({\mathfrak {g}})=U({\mathfrak

Hodge structure (4,864 words) [view diff] exact match in snippet view article find links to article

defining the hypersurface X {\displaystyle X} then the graded Jacobian quotient ring R ( f ) = C [ x 0 , … , x n + 1 ] ( ∂ f ∂ x 0 , … , ∂ f ∂ x n + 1 )

Glossary of commutative algebra (9,767 words) [view diff] exact match in snippet view article find links to article

submodule of a free module. total The total ring of fractions or total quotient ring of a ring is formed by forcing all non zero divisors to have inverses

Glossary of classical algebraic geometry (11,125 words) [view diff] exact match in snippet view article find links to article

and discussed by Dolgachev (2012, p.157). See also pippian. quotient ring The quotient ring of a point (or more generally a subvariety) is what is now