Commutative Algebra: Definitions

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Definitions

Undergrad

  • What is a prime ideal?
  • What is a primitive polynomial?
  • What is the content of a polynomial?
  • What is a separable extension
  • What is a Galois field extension?
    • What is a normal field extension?
    • What is a separable field extension?
  • What is a divisible group?
  • What are Ext and Tor?

Rings

  • What is the module-theoretic definition of a zero-divisor?

  • What is the annihilator \operatorname{Ann}(x) of an element x?

  • What is the ideal generated by a subset S of a ring R?

  • What is a reduced ring?

  • What is a local ring?

  • What is a regular ring?

  • What is a graded ring?

    • What is a filtered ring? What is the difference?
  • What is a Dedekind domain?

  • What is a DVR?

  • What is a Cohen-Macaulay ring?

  • What is a complete ring?

  • What is a normal ring?

  • What is a Noetherian ring?

  • What is a valuation ring?

  • What is a Jacobson ring?

  • What is an Artin ring?

  • What is a finite ring morphism?

  • What is a finite type morphism of rings?

  • What is a finitely generated ring?

  • What is a regular local ring? (Several characterizations)

  • What is a semilocal ring?

  • What is an excellent?

  • What is a Henselian ring?

  • What is a Gorenstein ring?

  • What is the localization of a ring?

    • What is a saturated multiplicative subset of a ring?
    • What are the p-local integers {\mathbf{Z}} \left[ { \scriptstyle { {p}^{-1}} } \right]?
  • What is the adic completion of a ring?

    • What are the p-complete integers {\mathbf{Z}}{ {}_{ \widehat{p} } }? What is the description as a set involving denominators?
  • What is a local property of a ring?

  • What is a flat morphism of rings?

  • What is a local morphism of local rings?

  • For R,S local rings, what does it mean for R to dominate S?

  • For a local ring (R, {\mathfrak{m}}), what is G_{\mathfrak{m}}(R)?

  • What is a connected ring?

  • What is \operatorname{Pic}(R) for R a ring.

Ideals

  • What is a homogeneous ideal?
  • What is the contraction (ideals) I_c of an ideal I?
  • What is the extension I^e of an ideal I?
  • What does it mean for two ideals to be coprime?
  • What is a fractional ideal?
  • What is the Unsorted/class group?
  • What does it mean for an ideal to belong to another ideal?
  • What is a factorial ring?
  • What is an irreducible ideal?
  • What is a nilpotent ideal?
  • What is a factorization of ideals?
  • What is a primary ideal?
  • What is the primary decomposition of an ideal?
  • What is the Rees algebra of an ideal?
  • What is an invertible ideal?
  • What is the group of ideals in a Dedekind domain?
  • For I{~\trianglelefteq~}R, what is the associated graded ring G_I(R)?
  • What is a complete set of orthogonal idempotents?

Modules and Algebras

  • What is an invertible module?
  • What is an algebra?
  • What is the tensor algebra of a module?
    • What is the alternating algebra of a module?
    • What is the symmetric algebra of a module?
  • What does it means for elements to generate an algebra?
  • What is an etale algebra?
  • What is a semisimple algebra?
  • What is a separable algebra?
  • What is a finitely generated A{\hbox{-}}algebra?
  • What is a finite A{\hbox{-}}algebra?
  • What is a finitely presented A{\hbox{-}}algebra?
  • What is a faithful module?
  • What is the radical of a submodule?
  • What is a primary submodule?
  • What does it mean for a module M to be module-finite over another module M'?
  • What is a finite type module M over a commutative algebra A?
  • What does it mean to be finitely generated as a module? As a field extension?
  • How does a ring morphism induce an algebra structure?
  • What is an integral A{\hbox{-}}algebra?
  • What is a reduced k{\hbox{-}}algebra?
  • What is the algebra structure on a tensor product of A{\hbox{-}}algebras?
  • What is base change or extension of scalars?
  • What is restriction of scalars?
  • What is extension of scalars?
  • What is the tensor product in {}_{R}{\mathsf{Mod}} ?
  • What is the tensor algebra in {}_{R}{\mathsf{Mod}} ?
  • What is the symmetric algebra in {}_{R}{\mathsf{Mod}} ?
  • What is the alternating algebra in {}_{R}{\mathsf{Mod}} ?
  • What is the exterior algebra in {}_{R}{\mathsf{Mod}} ?
  • What is a faithfully flat module (resp. morphism) in {}_{R}{\mathsf{Mod}} ?
  • What is a quotient morphism of modules?
  • What is a regular element of an algebra?
  • What is the Cayley-Hamilon theorem for modules?
  • What does it mean for a k{\hbox{-}}algebra to be formally smooth over k?

Dimension Theory and Numerics

  • What is the height of an ideal?
  • What is the depth of an ideal?
  • What is the codimension of an ideal?
  • What is the Krull dimension of a ring?
  • What is the global dimension of a ring?
  • What is the irrelevant ideal?
  • What is the length of a module?
  • What is an additive function on a class of modules?
  • What is the Poincare series of a graded module?
  • What is a Hilbert-Samuel polynomial?
  • What is a Hilbert function?
  • What is the characteristic polynomial of an {\mathfrak{m}}{\hbox{-}}primary ideal I?
  • What is a system of parameters?

AG

  • What is k[x]? k(x)? k[[x]]? k((x))?
  • What is the homogenization of a polynomial?
    • The dehomogenization?
  • For R a graded ring, what is \mathop{\mathrm{Proj}}R?
  • What is a constructible subset of a Noetherian space?
  • What is the coordinate ring A(V) for V an irreducible algebraic variety over k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu?
  • For k \coloneqq\operatorname{ff}(A), what is the rational function field k(V) on V?
  • What is the local dimension of an algebraic variety at a point?
  • What is a flat family?
  • What is faithfully flat base change?
  • What is a regular sequence?
  • What is the Plucker embedding?

Integrality

NT

  • What is a normal integral domain?
  • What is an algebraic number field?
  • What is the ring of integers?
  • What is a uniformizer?
  • What is a valuation?
  • What is the transcendence degree of a field extension?
  • For G\in {\mathsf{Ab}}{\mathsf{Grp}} totally ordered, what is the valuation ring of G?
    • What is the value group?
  • What is a Cauchy sequence in a topological group?
  • What is a monogenic field extension?
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