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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.

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Algebra over a ring

Could someone point me to a proof which shows that an algebra over a ring can be presented as a quotient of a polynomial ring (in possibly infinitely many variables).
user3714
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About the proof of Nullstellensatz

Let $k$ be an algebraically closed field. Let $\mathfrak{m}$ be a maximal ideal in $k[x_1,\ldots,x_n]$. Then $K:=k[x_1,\ldots,x_n]/\mathfrak{m}$ is a field. Moreover $K$ is also a finitely generated $k$-algebra, so by Zariski's Lemma $K$ is…
bateman
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About the multiplication map $I\otimes M\rightarrow M$

so I am learning about flat modules and I found this criterion for flatness. An $R$-module $M$ is flat iff for every finitely generated ideal $I$, we get that the multiplication map $I\otimes_R M\rightarrow M$ is an injection (Eisenbud Proposition…
Daniel Montealegre
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Dimension and Prime Filtrations

Question. Let $R$ be a Noetherian ring. Let $M$ be a finitely generated $R$-module such that $\dim(M)>0$. Take a prime filtration $0=K_0\subsetneq K_1\subsetneq ...\subsetneq K_t = M$, and let $P_i\in\text{Spec}(R)$ such that $K_i/K_{i-1}\cong…
Countable
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Counterexample of $f^* : \operatorname{Spec}B \to \operatorname{Spec}A$ injective implies every prime ideal of $B$ extended

This is exercise 3.20, (ii) of Atiyah & Macdonald. $f^*$ is the induced map on $\operatorname{Spec}$ of $f: A \to B$, a ring homomorphism. I have seen a counterexample on MathSE, stating that $k[t^2,t^3] \subset k[t]$ is one. But I cannot seem to…
Anthony Lee
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A ring is reduced if and only if it can be embedded into a product of fields

One of my homework questions asks me to prove that a commutative ring $A$ is reduced if and only if there exists fields $\{k_s\}_{s\in S}$ and an injective ring homomorphism $A\rightarrow \prod_{s\in S}k_s$. This is my attempt: Consider the…
Coco
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There is a bijection between $\operatorname{Hom}_{Ring}(A,\mathbb{F}_2)$ and $\operatorname{Spec} A$

Let $A$ be a Boolean ring. One of my homework problems asks me to prove that the map of sets $\operatorname{Hom}_{Ring}(A,\mathbb{F}_2)\to \operatorname{Spec} A$ defined by $$ \phi\mapsto \ker(\phi) $$ is a bijection. I have no idea how to do it. My…
Coco
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Why are there two different definitions of finitely generated rings in Atiyah-Macdonald?

Recently, I was confused by two different definitions in atiyah macdonald. In page 30, it says, A ring is said to be finitely generated if it is finitely generated as a $\mathbf Z$ algebra (i.e $A=\mathbf Z[a_1,a_2...a_n]$), but a ring is also an…
lee
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Proof of that integral closure of finitely generated domain $R$ over field is finitely generated $R$-module (Eisenbud, Corollary 13.13, Emmy Noether)

I am reading the Eisenbud, Commutative Algebra, Corollary 13.13 ( Emmy Noether ) and stuck at some statement. First, I propose a relavant question. Q. Let $R:=k[x_1 , \dots , x_d]$ with $\operatorname{char}(k)=p >0$ so that its fraction field is…
Plantation
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Kernel of $R$-module map $R^2\to R$, $(x,y)\mapsto ax+by$ for any $a,b\in R$

Let $R$ be an integral domain, and consider the $R$-module map $\phi \colon R^2\to R$ described by the matrix $\phi = (a,b)$ for some $a,b\in R$, meaning $$ \begin{pmatrix}a&b\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = ax+by $$ My question is…
Gargantuar
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Is it possible for a completion $\hat R$ to be finitely generated over $R$?

Let $(R,m)$ be a local Noetherian ring. Denote by $\hat R$ the $m$-adic completion of $R$. Is the following statement correct? If $\hat R$ is a finitely generated $R$-module, then $R=\hat R$.
Alex
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Proposition 5.15 from Atiyah-Macdonald how the previous lemma helps in concluding the coefficients are integral over the ideal

In chapter 5 of Atiyah Macdonald Lemma 5.14: Let $C$ is the integral closure of $A$ in $B$. Then the integral closure of ideal $\mathfrak{a}$ in $B$ is the $r(\mathfrak{a}^e)$ where $\mathfrak{a}^e$ is the extension of $\mathfrak{a}$ in…
Soham Chatterjee
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Why do the residue map extend to a place rather than an embedding?

Does anyone know how to fill the details of Lemma 2.3.4 from the book "Field Arithmetic" by M. D. Fried and M. Jarden. The statement of the lemma goes as follows: Let $v$ be a discrete valuation of a field $E$, $h\in O_v[X]$ a monic irreducible…
Jaakko Seppälä
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A homomorphism that extends to a place or an embedding

I tried to learn commutative algebra and I found a problem to do the following: Consider an integral domain $R$ and an algebraically closed field $M$. Let $\varphi_0$ be a homomorphism from $R$ to $M$, and let $F$ be a field that contains $R$. In…
Jaakko Seppälä
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What commutative Noetherian rings are locally principal?

It is well-known that the localization (at any multiplicative set) of any principal ideal ring is again a principal ideal ring (PIR). A Dedekind domain localized at any prime ideal is a DVR, so is again a PIR. These domains show that a commutative…
Chris Leary
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