Commutative Algebra: 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 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 ring?

  • 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 \({\mathbb{Z}} \left[ { \scriptstyle { {p}^{-1}} } \right]\)?

  • What is the adic completion of a ring?

    • What are the p-complete integers \({\mathbb{Z}}{ {}_{ \widehat{p} } }\)? 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 \(I_c\) 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 ideal 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 \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)?
  • What is the tensor algebra in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)?
  • What is the symmetric algebra in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)?
  • What is the alternating algebra in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)?
  • What is a faithfully flat module (resp. morphism) in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)?
  • What is a quotient morphism of modules?
  • What is a regular element of an algebra?
  • What are the norm and trace of a module?
  • 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

Integrality

  • What is an integral extension of rings?
  • What is the integral closure of a subring?
  • What does it mean for an integral domain to be ntegrally closed?

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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