Tags: #studyguides
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 moduletheoretic definition of a zerodivisor?

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 CohenMacaulay 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 plocal integers \({\mathbf{Z}} \left[ { \scriptstyle { {p}^{1}} } \right]\)?

What is the adic completion of a ring?
 What are the pcomplete 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 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?

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 modulefinite 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 CayleyHamilon 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 HilbertSamuel 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 \(k=\mkern 1.5mu\overline{\mkern1.5muk\mkern1.5mu}\mkern 1.5mu\)?
 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?

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?