requires non-trivial generalization of topological spaces as "non-commutative spaces". For this reason there is some talk about non-commutative topology...

arbitrary commutative ring as a foundation for algebraic geometry. The theory took its definitive form in Grothendieck's Éléments de géométrie algébrique...

Mathematics (N)". Maths History. Fontenelle, Bernard de (1727). Eléments de la géométrie de l'infini (in French). p. 3. Arithmetices principia: nova methodo (in...

The Éléments de géométrie algébrique (EGA; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a...

(5): 433–439, doi:10.1512/iumj.1974.23.23036 Connes, A. (1990), Géométrie non commutative, InterEditions, ISBN 2-7296-0284-4 Ferrara, S.; Mattiolia, G.;...

to be commutative when the diagram commutes for all objects A {\displaystyle A} and B {\displaystyle B} . The Kleisli category of a commutative monad...

and C*-algebras are known as non-commutative measure theory and non-commutative topology, respectively. Non-commutative geometry is not merely a pursuit...

Flatness was introduced by Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. A left module M over a ring R is flat if the following...

originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings...

This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic...

In algebraic geometry and commutative algebra, a ring homomorphism f : A → B {\displaystyle f:A\to B} is called formally smooth (from French: Formellement...

the structure of quasi-finite morphisms of schemes. Several results in commutative algebra imply the geometric forms of Zariski's main theorem, including:...

Algébriques Cohérents (FAC, 1955), on coherent cohomology, and Géométrie Algébrique et Géométrie Analytique (GAGA, 1956). Even at an early stage in his work...

notion of a complete intersection. Given a commutative ring R and an R-module M, an element r in R is called a non-zero-divisor on M if r m = 0 implies m...

that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring...

multiplication is non-commutative). This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative...

such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case...

Pontryagin duality taking finite commutative group schemes to finite commutative group schemes. Finite flat commutative group schemes over a perfect field...

Grothendieck, J. Dieudonné, Eléments de géométrie algébrique, Ch. 0IV § 23, Publ. Math. IHÉS 20, (1964). H. Matsumura, Commutative algebra ISBN 0-8053-7026-9, chapter...

of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros...

contents of polynomials". In Chapman, Scott T.; Glaz, Sarah (eds.). Non-Noetherian Commutative Ring Theory. Mathematics and its Application. Vol. 520. Dordrecht:...

of the subject. For the items in commutative algebra (the theory of commutative rings), see Glossary of commutative algebra. For ring-theoretic concepts...

1016/0040-9383(90)90003-3, Zbl 0759.58047 Skandalis, Georges (2001). "Géométrie non commutative, opérateur de signature transverse et algèbres de Hopf (d'après...

rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative...

In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules...

first of these comes from commutative algebra: subject to some finiteness conditions on f, it can be shown that there is a non-empty open subscheme Y ′...

Outside of algebraic geometry, local cohomology has found applications in commutative algebra, combinatorics, and certain kinds of partial differential equations...

III.2 Porst, Hans-E. (1981), "Characterization of injective envelopes", Cahiers de Topologie et Géométrie Différentielle Catégoriques, 22 (4): 399–406...

Banach–Stone, and allows one to consider noncommutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of noncommutative...

(sometimes ^^ instead of ^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation. For nonnegative integers n and...