Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

16857 questions
3
votes
1 answer

Poincare Series and Hilbert Polynomial of graded $S$-modules

I am trying to find the Poincare series and Hilbert polynomial for graded $S$-modules $I=S \cdot T^m$ and $M=S/I$ where $S=k[T]$ is the graded polynomial algebra and $m \geq 1$. I am not particularly comfortable with this material, so I was…
user 3462
  • 1,493
  • 7
  • 16
3
votes
4 answers

Radical ideal of $(x,y^2)$

How does one show that the radical of $(x,y^2)$ is $(x,y)$ over $\mathbb{Q}[x,y]$? I have no idea how to do, please help me.
Jr.
  • 4,026
3
votes
1 answer

How to find a generating set for the lattice ideal when one has a lattice basis?

This is an example from Miller, Sturmfels "Combinatorial Commutative Algebra". Let $\pi:\mathbb{Z}^3 \rightarrow \mathbb{Z}$ be the group homomorphism defined by the matrix $\left(\begin{array}{ccc} 3 & 4 & 5 \end{array} \right)$. Thus…
RedOrange
  • 321
  • 2
  • 9
3
votes
1 answer

Primary decomposition in a quotient ring

Suppose $I$ and $J$ are two ideals in a polynomial ring $R=\mathbb{Q}[x_1,\dots,x_n]$, what's the relation between the primary decomposition of $I$ in the quotient ring $R/J$ and the primary decomposition of $I+J$ in $R$? In particular it can be…
l'etranger
  • 33
  • 2
3
votes
1 answer

That submodule generated by one element leads to submodule being finitely generated

In Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, in the prove of Proposition 1.4, the auther seems to use the following fact. Let $R$ be a Noetherian ring, $M$ is a finitely generated $R$-module, $N$ is a submodule of…
hxhxhx88
  • 5,257
3
votes
3 answers

Example of invertible maximal ideal that is not generated by one element

Could anyone give me an example of an invertible maximal ideal of some integral domain which is not generated by one element?
hxhxhx88
  • 5,257
3
votes
1 answer

Prove that in a Noetherian ring, no invertible maximal ideal properly contains a nonzero prime ideal

Let $R$ be an integral domain which is Noetherian, let $P$ be an invertible maximal ideal, and let $Q
hxhxhx88
  • 5,257
3
votes
1 answer

Every set of $n$ generators of $A^{n}$ is actually a basis

Let $A$ be a commutative ring with $1$. It is a standard result that every set of $n$ generators of the free $A$-module $A^{n}$ is actually a basis. The proof uses tensor products. I was reading a "proof" of this argument: Let $\{m_{1},..,m_{t}\}$…
user6495
  • 3,957
3
votes
1 answer

Can two non-isomorphic local domains have isomorphic quotient field and residue field?

Suppose $R,R'$ are non-isomorphic local domains with maximal ideals $\mathfrak{m,m'}$. Can they have isomorphic quotient field and residue field?
user93417
3
votes
1 answer

isomorphic localization of finitely presented modules

Let $A$ be commutative ring with unit and M,N be two finitely presented modules over $A$. I want to show that if $M_{\mathfrak{p}} \cong N_{\mathfrak{p}}$ for some prime ideal $\mathfrak{p}$ of $A$ then there exists $f \in A \backslash \mathfrak{p}$…
Zorba le Grec
  • 999
  • 6
  • 17
3
votes
1 answer

Generating set of a free module is its basis

Let $F \cong R^d$ be a free R-module, where R is Noetherian. Let $Y$ be a generating set for $F$ with $|Y| \leqslant d$. Show that $Y$ is a basis for $F$ and $|Y|=d$. Thought that this must be like that, bug got the outcome "prove it!".
AlexCon
  • 745
3
votes
1 answer

Which ideals are radical?

For a commutative ring $R$ and ideal $A$, let $N(A)=\{x \in R\mid $ there exists a nonnegative integer $n$ such that $x^n \in A\}$. For which of the following $R$ and $A$ is it true that $N(A)=A$ ? I. $R=\Bbb Z,\ A=(2)$ II. $R=\Bbb Z[x],\…
user101204
  • 111
3
votes
1 answer

A subring of polynomial ring with coefficients in a DVR that is not noetherian

Let $R$ be a discrete valuation ring, $K$ its field of fractions and $A=\{f\in K[T],f(0)\in R\}$. Let $\mathfrak{m}$ be the maximal ideal of $R$, $\mathfrak{m'}={\mathfrak{m}+KT+KT^2+\cdots}$. 1) How to show $A$ is not a Noetherian ring? I cannot…
user93417
3
votes
0 answers

Completion of intersection of prime ideals

Let $R$ denote the ring of convergent power series over $\mathbb{C}$ in $n$ variables (which is a Noetherian, excellent, local ring). For any finite set of prime ideals one has that the $\mathfrak{m}$-adic closure commutes with taking intersections,…
Sebastian
  • 1,302
  • 7
  • 16
3
votes
1 answer

homomorphism of graded modules

This is a little exercise problem from Peeva's book on graded syzygies. Let $\phi:N\to T$ be a homomorphism of graded $R$-modules. If $f=f_1+\cdots+f_n\in N$ and $f_i$ are its homogeneous components, then $\phi(f_i)$ are homogeneous components of…
D. Huang
  • 579
1 2 3
…
99 100