Tags: #study-guides
Definitions
Undergrad
- What is a prime ideal?
- What is a primitive polynomial?
- What is the content of a polynomial?
- What is a separable extension
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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
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What is the module-theoretic definition of a zero-divisor?
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What is the annihilator \operatorname{Ann}(x) of an element x?
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What is the ideal generated by a subset S of a ring R?
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What is a reduced ring?
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What is a local ring?
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What is a regular ring?
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What is a graded ring?
- What is a filtered ring? What is the difference?
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What is a Dedekind domain?
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What is a DVR?
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What is a Cohen-Macaulay ring?
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What is a complete ring?
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What is a normal ring?
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What is a Noetherian ring?
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What is a valuation ring?
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What is a Jacobson ring?
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What is an Artin ring?
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What is a finite ring morphism?
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What is a finite type morphism of rings?
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What is a finitely generated ring?
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What is a regular local ring? (Several characterizations)
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What is a semilocal ring?
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What is an excellent?
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What is a Henselian ring?
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What is a Gorenstein ring?
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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]?
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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?
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What is a local property of a ring?
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What is a flat morphism of rings?
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What is a local morphism of local rings?
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For R,S local rings, what does it mean for R to dominate S?
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For a local ring (R, {\mathfrak{m}}), what is G_{\mathfrak{m}}(R)?
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What is a connected ring?
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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?
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What is a fractional ideal?
- What is a principal fractional ideal?
- What is a colon ideal?
- What does it mean for a fractional ideal to be invertible?
- 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?
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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))?
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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
- 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 integrally 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?
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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?