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

searching for Commutative ring 132 found (597 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

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

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

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

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

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"

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

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

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

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

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

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

commutative algebra and algebraic geometry, a module over a Noetherian commutative ring R {\displaystyle R} can have finite length only when the module has

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

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

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

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

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

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

] {\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 ] / t n

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

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

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

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

"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

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

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

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

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

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   →

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

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

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

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

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

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

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

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;

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.

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

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

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

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

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

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

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

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

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

1007/978-1-4612-5350-1. ISBN 0-387-94268-8. MR 1322960. Matsumura, Hideyuki (1989). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated

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 L n

representation amounts to choosing an identity for the heap. Let R be a commutative ring with identity, and let R-Mod be the category of R-modules. If M and

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

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

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

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

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

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

Math. 86 (1): 93–98. doi:10.2140/pjm.1980.86.93. —— (1994). "A quasi-commutative ring that is not neo-commutative". Proc. Amer. Math. Soc. 122: 321. doi:10

Math. 86 (1): 93–98. doi:10.2140/pjm.1980.86.93. —— (1994). "A quasi-commutative ring that is not neo-commutative". Proc. Amer. Math. Soc. 122: 321. doi:10

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

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

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

\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

 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

R ) ≅ 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

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

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

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

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

{\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

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

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

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

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

\\\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

{\displaystyle p\colon E\to B} with fiber F {\displaystyle F} and a fixed commutative ring R {\displaystyle R} with a unit, there exists a contravariant functor

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

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

algebra of the pair (G, K is the classical Hecke algebra, which is the commutative ring of Hecke operators in the theory of modular forms. The case leading

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

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

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

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

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

ideals a , b {\displaystyle {\mathfrak {a}},{\mathfrak {b}}} of a commutative ring R, the R-annihilator of ( b + a ) / a {\displaystyle ({\mathfrak {b}}+{\mathfrak

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

|x|\leq 1} hold. More generally, if A {\displaystyle A} is a topological commutative ring whose topology is induced by an absolute value, then an element x ∈

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

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

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

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

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

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

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.

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

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

{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

In a commutative ring, an equivalence class modulo squares

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

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

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]}

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

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

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

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

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

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

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

Segal–Bargmann transform. The Heisenberg group Hn(K) is defined for any commutative ring K. In this section let us specialize to the field K = Z/pZ for p a

scheme, a locally-ringed space that is isomorphic to the spectrum of a commutative ring. Each affine variety has an affine scheme associated to it: if V(I)

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

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}

result has been generalized still further, e.g. by working over any commutative ring in which 2 is cancellable, meaning that 2x = 2y implies x = y. Composition

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

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