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Longer titles found: Commutative ring spectrum (view), Noncommutative ring (view), Graded-commutative ring (view), Simplicial commutative ring (view)

searching for Commutative ring 134 found (590 total)

alternate case: commutative ring

Nilpotent algebra (388 words) [view diff] exact match in snippet view article find links to article

specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product

space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection

30.2 and Theorem 30.3 in: Matsumura, Commutative Ring Theory (1989). Theorem 30.3, Matsumura, Commutative Ring Theory (1989). Lemma 1 in section 28 and

(resp. finitely presented). For finitely generated modules over a commutative ring R, Nakayama's lemma is fundamental. Sometimes, the lemma allows one

ideals. Noetherian domains and Krull domains both have this property. A commutative ring is a Krull domain if and only if it is a Mori domain and completely

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is

(specifically commutative ring theory), Faltings' annihilator theorem states: given a finitely generated module M over a Noetherian commutative ring A and ideals

an operad. It is a generalization of an associative algebra over a commutative ring R, with an operad replacing R. Given an operad O (say, a symmetric

geometry, a derived stack is, roughly, a stack together with a sheaf of commutative ring spectra. It generalizes a derived scheme. Derived stacks are the "spaces"

subsets of X, where morphisms are inclusions, to the good category of commutative ring spectra. A theorem of Jardine says that such presheaves form a simplicial

In mathematics, a Marot ring, introduced by Marot (1969), is a commutative ring whose regular ideals are generated by regular elements. Marot, Jean (1969)

still see modern usage, for example, in the obstruction theory of commutative ring spectra. Examples of secondary and higher cohomology operations include

In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning

multiplicative sets include: the set-theoretic complement of a prime ideal in a commutative ring; the set {1, x, x2, x3, ...}, where x is an element of a ring; the

called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product [ _ , _ ] satisfying the Leibniz identity

dual Steenrod algebra A ∗ {\displaystyle {\mathcal {A}}^{*}} is a commutative ring, and the setting of commutative algebra provides more tools for studying

In algebra, the integral closure of an ideal I of a commutative ring R, denoted by I¯{\displaystyle {\overline {I}}}, is the set of all elements r in R

over a commutative ring is that a bilinear form is unimodular if V → V∗ is an isomorphism. Given a finitely generated module over a commutative ring, the

projective module over a local ring is free; where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit

of the universal coefficient theorem. Let R {\displaystyle R} be a commutative ring. For R {\displaystyle R} -orientable M {\displaystyle M} with fundamental

ring. Tate algebras over non-archimedean fields are Jacobson rings. A commutative ring R is a Jacobson ring if and only if R[x], the ring of polynomials over

Ranthony A. C. Edmonds is an American mathematician specializing in commutative ring theory, factorization theory, and applied algebraic topology. She is

"Subalgebra" can refer to either case. A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication

Since the product of two regular elements (=non-zerodivisors) of a commutative ring R is again a regular element, it is apparent that the product of two

R[[t]]{\displaystyle \textstyle R[[t]]} of formal power series over a commutative ring R can be thought of as the inverse limit of the rings R[t]/tnR[t]{\displaystyle

submodule determinant The determinant of a finite free module over a commutative ring is the r-th exterior power of the module when r is the rank of the

sections of E → X {\displaystyle E\to X} . Let A {\displaystyle A} be a commutative ring and M {\displaystyle M} an A-module. There are different equivalent

intersection of primary ideals. Let R {\displaystyle R} be a Noetherian commutative ring. An ideal I {\displaystyle I} of R {\displaystyle R} is called primary

detected from the local rings. For instance, being a flat module over a commutative ring is a local property, but being a free module is not. For more, see

"weak inverse" of x. The following statements are equivalent for the commutative ring R: R is von Neumann regular. R has Krull dimension 0 and is reduced

GL2(Q) and K = GL2(Z) we have the classical Hecke algebra, which is the commutative ring of Hecke operators in the theory of modular forms. The case leading

R)\cong k;} R is an n-dimensional Gorenstein ring. A (not necessarily commutative) ring R is called Gorenstein if R has finite injective dimension both as

elucidate the multiplicative properties of necklace polynomials. If A is a commutative ring then the necklace ring over A consists of all infinite sequences (

points of access to the basic definitions. An Azumaya algebra over a commutative ring R{\displaystyle R} is an R{\displaystyle R}-algebra A{\displaystyle

module, the relations between these relations, and so forth. Let A be a commutative ring and s: Ar → A an A-linear map. Its Koszul complex Ks is ∧ r A r   →

representations of the symmetric group on n points. Fix a partition λ of n and a commutative ring k. The partition determines a Young diagram with n boxes. A Young tableau

are necessary and sufficient. If B is a faithfully flat commutative ring over the commutative ring A, then the functor T from A modules to B modules taking

operad. The Koszul-dual of Lie{\displaystyle {\mathcal {Lie}}} is the commutative-ring operad, an operad whose algebras are the commutative rings over k.

1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute

{\displaystyle \operatorname {Ext} ^{*}(\cdot ,\cdot )} groups. Given a commutative ring R {\displaystyle R} and a module M {\displaystyle M} , the Yoneda product

r in the ring, then adding r to itself n times gives nr = 0. If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all

submodules Irreducible ideal Irreducible representation Herstein, Non-commutative Ring Theory, Lemma 1.1.3 Serre, Jean-Pierre (1977). Linear Representations

}g_{\mu ,\nu }^{\lambda }(q)u_{\lambda }.\,} It turns out that H is a commutative ring, freely generated by the elements u1n{\displaystyle u_{\mathbf {1}

This is because the structure of an A {\displaystyle A} -algebra on a commutative ring B {\displaystyle B} is directly encoded by a ring morphism A → B {\displaystyle

1977 (ISBN 978-3642380099) And he has also translated Matsumura's Commutative Ring Theory, Cambridge University Press 1987, doi:10.1017/CBO9781139171762

doi:10.2140/pjm.1958.8.511, ISSN 0030-8730, MR 0099360 Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8. Mitchell

of co-cones from F. In the category ChR of chain complexes over a commutative ring R, the zero complex is a zero object. In a short exact sequence of

of commutative k{\displaystyle k}-algebras. A Hopf algebroid over a commutative ring k{\displaystyle k} is a pair of k{\displaystyle k}-algebras (A,Γ){\displaystyle

a CW-complex K(Z,n) called an Eilenberg–MacLane space. Let R be a commutative ring with identity, and let R-Mod be the category of R-modules. If M and

similar to the dimension of vector spaces. The Krull dimension of a commutative ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal

q}^{r}} be first quadrant spectral sequences of flat modules over a commutative ring and f:Er→′Er{\displaystyle f:E^{r}\to {}^{\prime }E^{r}} a morphism

Clifford algebra. Witt vector, an infinite sequence of elements of a commutative ring, which belongs to an algebra over this ring, and has been introduced

following construction is common: given an ideal I {\displaystyle I} in a commutative ring R , {\displaystyle R,} the I-adic topology on R {\displaystyle R} is

Abstract linear algebra considers matrices with entries from any commutative ring R{\displaystyle R}, not limited to the integers. In this context, a

integral homology vanishes. The second algebraic K-group K2(R) of a commutative ring R can be identified with the second homology group H2(E(R), Z) of the

and modules, for which equality holds. Let R {\displaystyle R} be a commutative ring, I {\displaystyle I} an ideal of R {\displaystyle R} and M {\displaystyle

Abstract Nash functions can also be defined on the real spectrum of any commutative ring. J. Bochnak, M. Coste and M-F. Roy: Real algebraic geometry. Springer

2018. Harrison, D. K.; Vitulli, M. A. (1989). "V-Valuations of a Commutative Ring I". J. Algebra. 65 (3): 264–292. doi:10.1016/0021-8693(89)90305-0.

ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 Matsumura, Hideyuki (2003), Commutative ring theory, Translated from the Japanese by M. Reid. Cambridge Studies

definition of the cohomology of the dual of a chain complex. Let R be a commutative ring, C a chain complex of R-modules and G an R-module. Just as one lets

assertions in this article appear in (Lam 1999) and (Lam 2001). For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the

Matsumura 1989, pg. 215 Matsumura 1989, Theorem 28.7 Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid

geometric applications Lang 2002, p. 142 "Let E be a free module over a commutative ring A. We view A as a free module of rank 1 over itself. By the dual module

intersection ring Fröberg, R. (1999), "Koszul algebras", Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Applied Mathematics,

Grégoire, Benjamin; Mahboubi, Assia (2005). "Proving Equalities in a Commutative Ring Done Right in Coq". In Hurd, Joe; Melham, Tom (eds.). Theorem Proving

CiteSeerX 10.1.1.568.9148, doi:10.2307/120991, JSTOR 120991 Moduli spaces of commutative ring spectra (PDF) Goerss – Topological Modular Forms (PDF) Ando, Matthew;

space according to a very ample L and its global sections. The graded commutative ring that is formed by the direct sum of the global sections of the Ln, {\displaystyle

complexes of K-modules with chain maps form a category ChK, where K is a commutative ring. If V = V ∗ {\displaystyle {}_{*}} and W = W ∗ {\displaystyle {}_{*}}

the idealizer is related to a more general construction. Given a commutative ring R, and given two subsets A and B of a right R-module M, the conductor

gives conditions one can check to show flatness of a module. Given a commutative ring A, an ideal I and an A-module M, suppose either A is a Noetherian ring

Epsilon Journal. In 1942, McCoy proved the following theorem: Let R be a commutative ring with multiplicative unit, and let f(x) be a zero divisor in the polynomial

\mathbb {P} _{A}^{n}} be the n-dimensional projective space over a commutative ring A. Let Ω 1 = Ω P A n / A 1 {\displaystyle \Omega ^{1}=\Omega _{\mathbb

sends ∗ {\displaystyle \ast } to 1. If R {\displaystyle R} is a (commutative) ring, then the free functor S e t , → R − m o d {\displaystyle {\mathsf

Polynomials on a Subset?". In Hazewinkel, Michiel (ed.). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. Vol. 520. Springer. p. 85

{\displaystyle 2} after finitely many objects in the resolution. For a commutative ring S {\displaystyle S} and an element f ∈ S {\displaystyle f\in S} , a

\\\end{aligned}}} and all the other homology groups are zero. For a general commutative ring R, the homology of X and Y is related to the homology of their product

for Noetherian schemes: First, given a projective module E over a commutative ring A, an A-linear map u : E → A {\displaystyle u:E\to A} is called Koszul-regular

modules soc(RR)≅R/J{\displaystyle \mathrm {soc} (_{R}R)\cong R/J}. For a commutative ring R, the following are equivalent: R is Frobenius R is quasi-Frobenius

far-reaching generalization of Goodwillie's result, stating that for a commutative ring A so that the Henselian lemma holds with respect to the ideal I, the

Weierstrass division and preparation, with A being a not necessarily commutative ring, and with formal skew power series in place of formal power series

{\displaystyle H^{*}(X,O_{X})=\bigoplus _{j}H^{j}(X,O_{X})} into a graded-commutative ring, meaning that v u = ( − 1 ) i j u v {\displaystyle vu=(-1)^{ij}uv}

linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split

its earliest stages, it was known that K could be replaced by any commutative ring, provided that L is a free K-module, i.e., has a basis as above. To

ISBN 978-3-642-27436-7. Spear, David (July 1977). "A constructive approach to commutative ring theory". Proceedings of the 1977 MACSYMA Users' Conference. Mora, Teo

the set of finite type schemes over a field k{\displaystyle k} to a commutative ring A{\displaystyle A}, satisfying the three properties μ(X){\displaystyle

Artin–Wedderburn theorem. For commutative rings, several areas together led to commutative ring theory. In two papers in 1828 and 1832, Gauss formulated the Gaussian

ISBN 978-1-4704-1884-7. Goerss, P. and Hopkins, M., Moduli Spaces of Commutative Ring Spectra, http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf Hopkins

no. 4, 857–892. Goerss, P. G., and M. J. Hopkins. "Moduli spaces of commutative ring spectra." Structured ring spectra, 151–200, London Math. Soc. Lecture

{\displaystyle \mathbb {C} .} Since A {\displaystyle A} is then a commutative ring with unit, every non-invertible element of A {\displaystyle A} belongs

Hochster's direct summand conjecture that any finite extension of a regular commutative ring splits as a module. In 2011, André received the Prix Paul Doistau–Émile

The Hadamard product is commutative (when working with a commutative ring), associative and distributive over addition. That is, if A, B, and

commutative, associative, and unital. If we let A {\displaystyle A} be a commutative ring, and let A ∖ R {\displaystyle A\backslash R} be the undercategory of

{torsion} } be the second homology modulo its torsion. Let R be any commutative ring with unit and Λ the ring of formal power series of the form λ = ∑ A

connected. More generally, the following are equivalent The spectrum of a commutative ring R {\displaystyle R} is connected Every finitely generated projective

series in the variable x{\displaystyle x}, with coefficients in a commutative ring R{\displaystyle R}. Denote by R0[[x]]⊂R[[x]]{\displaystyle R^{0}[[x]]\subset

Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring", Revue Roumaine de Mathématiques Pures et Appliquées, 43: 591–603

) Then for f , g : P → K {\displaystyle f,g:P\to K} , where K is a commutative ring, we have g ( t ) = ∑ s ≤ t f ( s )  for all  t ∈ P {\displaystyle g(t)=\sum

 390–405 doi:10.2307/1992937 with Oscar Goldman: The Brauer group of a commutative ring, Trans. Amer. Math. Soc., vol. 97, no. 3, 1960, pp. 367–409 doi:10

Auslander, Maurice; Goldman, Oscar (1960). "The Brauer group of a commutative ring". Transactions of the American Mathematical Society. 97 (3): 367–409

groups. Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra over a commutative ring R with universal enveloping algebra U g {\displaystyle U{\mathfrak

all j ≥ i and is injective for all j ≥ i − 1. For any field k and commutative ring R, Voevodsky defined an R-linear triangulated category called the derived

93–98. doi:10.2140/pjm.1980.86.93. Kaplansky, Irving (1994). "A quasi-commutative ring that is not neo-commutative". Proc. Amer. Math. Soc. 122: 321. doi:10

equal unity) for weighted digraphs with arc weights taken from a given commutative ring. In the meantime, for an undirected weighted graph the sum of the products

left nil ideal is in the nilradical. An element of a (possibly non-commutative ring) is called left singular if it annihilates an essential left ideal

cohomology of a Lie algebra g {\displaystyle {\mathfrak {g}}} over some commutative ring k {\displaystyle k} is the right derived functor of the invariants

group is the additive group of L. Define A to be Fp[T], the usual (commutative!) ring of polynomials over the finite field of order p. In other words, A

In a commutative ring, an equivalence class modulo squares

Hochschild cohomology (Shukla cohomology): Let k {\displaystyle k} be a commutative ring, and let A {\displaystyle A} be a commutative k − {\displaystyle k-}

Yang–Baxter equation and mixable shuffle product constructions. Let k be a commutative ring and let λ{\displaystyle \lambda } be given. A linear operator R on

polynomials over any commutative ring R, and have the following property. Let φ be a ring homomorphism of R into another commutative ring S. It extends to

linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split

limits. A ringed topos is a pair (X,R), where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through

Michel Coste and Marie-Françoise Roy discover the real spectrum of a commutative ring. 1980 Oleg Viro introduced the "patch working" technique and used it

Teacher, Vol. 81, 1988, pp. 583–592 A sketch of the evolution of (non-commutative) ring theory, L´Enseignement Mathematique, Vol. 33, 1987, pp. 227–267 The

adjunction expresses a universal property of group rings. Let R be a (commutative) ring, let G be a group, and let S be an R-algebra. For any group homomorphism

Let M{\displaystyle M} be an affine monoid, and R{\displaystyle R} a commutative ring. Then one can form the affine monoid ring R[M]{\displaystyle R[M]}

adjunction expresses a universal property of group rings. Let R be a (commutative) ring, let G be a group, and let S be an R-algebra. For any group homomorphism

constant sheaf of rings A_{\displaystyle {\underline {A}}} for a fixed commutative ring A{\displaystyle A}. Then, the sheaves will be sheaves of A_{\displaystyle

it is an automorphism, but this is not true in general. Let R be a commutative ring with prime characteristic p (an integral domain of positive characteristic

scheme, a locally-ringed space that is isomorphic to the spectrum of a commutative ring (up to an equivalence of categories). Each affine variety has an affine

stable homotopy group of spheres. This makes it possible to take a commutative ring A {\displaystyle A} and constructing a complex analogous to the Hochschild

groupoid of X {\displaystyle X} to the category of modules over a commutative ring R {\displaystyle R} , where typically R = Q , R , C {\displaystyle

advisory of mathematician Yasuo Akizuki. Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. ISBN 9781139171762.

S) is a Coxeter system with the Coxeter matrix M = (mst), R is a commutative ring with identity. {qs | s ∈ S} is a family of units of R such that qs

2020. Harrison, D. K.; Vitulli, Marie A. (1989). "V-Valuations of a Commutative Ring I". J. Algebra. 65 (3): 264–292. doi:10.1016/0021-8693(89)90305-0.

high-level way in terms of algebra isomorphisms. We first recall that for a commutative ring R{\displaystyle R} and a group isomorphism from G{\displaystyle G}

define his version of perfect ideals. Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 132. ISBN 9781139171762

linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split

{grade}}_{R}\,I={\textrm {depth}}_{I}(R).} Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 131. ISBN 9781139171762

algebra functor on the category of R {\displaystyle R} -modules for a commutative ring R {\displaystyle R} . Sym ∙ ( − ) : Mod ( R ) → Mod ( R ) {\displaystyle

MR 1545800, S2CID 121374449, Zbl 0020.34003 Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Translated

k {\displaystyle E_{k}} or D k {\displaystyle D_{k}} , is the non-commutative ring over W ( k ) {\displaystyle W(k)} generated by 2 elements F {\displaystyle

be an n×n{\displaystyle n\times n} matrix over a (not necessarily commutative) ring R{\displaystyle R} and fix 1≤i,j≤n{\displaystyle 1\leq i,j\leq n}

Enderton p. 104 and sketched for a general field of fractions over a commutative ring in Dummit and Foote p. 263. Enderton p. 114 Ferreirós p. 135; see section