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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
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Subsequences of regular sequence

I was trying to answer this question - whether a subsequence of a regular sequence is regular in a Noetherian ring which is not local. In the local case, regular sequences can be permuted and so a subsequence can be considered to be the initial…
Dev Bappa
  • 419
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Noetherian module over noetherian ring

Let $M$ be a noetherian module over noetherian ring $A$. How to prove that there exists submodule $N\subset M$ such that $$M/N\cong A/\mathfrak{p}$$ for some prime ideal $\mathfrak{p}\in A$. Is it true that any submodule of noetherian module over…
Aspirin
  • 5,659
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$M/IM=0$ for all maximal $I \Longleftrightarrow M=0$?

Let $A$ to be a commutative ring with an identity $1$. $M$ is a f.g. $A$--module, Does $$M/IM=0 \text{ for all maximal } I \Longleftrightarrow M=0$$ hold? It seems like that using localization could solve this (not sure). Is there any other way to…
Lwins
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$B$ is a flat $A$-algebra. Does $IB \cap JB = (I \cap J)B$ holds?

$B$ is a flat $A$-algebra. For any ideals $I, J \in A$, does $IB \cap JB = (I \cap J)B$ holds? (The answer seems to be YES, how to prove it?) Furthermore, if $f: A \to B$ is injective (maybe $B$ is not a flat $A$-module), does the equality still…
Lwins
  • 624
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why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?

the question is exactly "why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?" Is this simply because in the normalization process we can have many irreducible component?
unknown
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Confusion about the properties ''regular'' and ''$\exists$ a system of parameters'' for a local ring.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of Krull-dimension $d$ with maximal ideal $\mathfrak{m}$. By definition a system of parameters are elements $a_1,\ldots, a_d\in \mathfrak{m}$ with $$ \mathfrak{m}=\sqrt{(a_1,\ldots, a_d)}.…
7832468
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Equivalent definitions of associated prime ideals

I'm reading the book Undergraduate Commutative Algebra of M. Reid. He gives the following definition: Let $M$ be an $R$-module and let $P$ be an ideal of $R$. Then $P$ is an associated prime if: $1.$ $ P\in \text{Spec}(R)$ $2.$ there exists $x…
harajm
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Characteristic of a ring $A$ and residue fields

Let $p$ be a prime number and $A$ be a commutative ring with unity. We say that $A$ has characteristic $p$ if $p\cdot 1_A=0$. I would like to know if you could have a ring $A$ with all residue fields (= $\operatorname{Frac}(A/\mathfrak{p}$) with…
Abellan
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I want to have example of a ring extension where lying over holds but going up does not.

Let $A\subset B$ be a ring extension where every prime ideal of $A$ is contraction of a prime ideal in $B$. We have to find prime ideals $P_1\subset P_2$ in $A$ and a prime ideal $Q_1$ in $B$ such that $P_1=Q_1\cap A$ and there is no prime ideal…
Duster
  • 59
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Completion of a polynomial ring w.r.t the homogeneous maximal ideal is the power series ring

I am trying to understand the proof of this fact. On page 183, Eisenbud defines a map from the formal power series ring $S[[x_1,x_2,...,x_n]]$ to $R/m^i$ where $R=S[x_1,x_2,...,x_n]$ sending $f$ to $f+m^i$. I have trouble understanding this map…
Dev Bappa
  • 419
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Behaviour of conductor ideal

Let $f : A \to B$ be a homomorphism of finitely generated $k$-algebras, where $k$ is a field. Let $J_A$ and $J_B$ denote the conductor ideals of $A$ and $B$ respectively for the corresponding normalizations in the quotient fields (assume that $A,…
Math Blogger
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Ring theoretic properties of integer valued poynomials.

Let us denote by $A$ the ring of integer-valued polynomials in $\mathbf{Q}[T]$. We know that $A$ is not Noetherian and of dimension $2$. I would like to understand $A$ better, for instance (a) what is the homological dimension of $A$? (b) I think…
David Gilbert
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showing $k[x,y]\ncong k[u,v,w]/(uw-v^2)$

Let $k$ be a field. I want to show that $k[x,y]\ncong k[u,v,w]/(uw-v^2)$ as $k$-algebras, but can't find a way to do it. The dimension of the $k$-vector space generated by the degree 1 monomials are different on both sides, but then it's possible…
Hajime_Saito
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Show that if $f_{M}: L_{M} \to G_{M}$ is surjective for every maximal ideal $ M$ of $R$ then $f$ is surjective.

Let $f:L\to G$ be a homomorphism of modules over commutative ring $R$. Show that if $f_{M}: L_{M} \to G_{M}$ is surjective for every maximal ideal $ M$ of $R$ then $f$ is surjective.
Thế Long Lê
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What is the tensor product $M_n(L)\otimes_K L$, where $L/K$ is a quadratic extension?

What is the tensor product $M_n(L)\otimes_K L$, where $L/K$ is a quadratic extension? Let $K$ be a field of characteristic $0$, $L/K$ a quadratic extension. Let $\rho\in \operatorname{Gal}(L/K)$ denote the nontrivial element of the Galois group. Let…
Mikhail Borovoi
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