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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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Fraction field of $F[x,y,z]/(x^4+y^4+zy)$ equal to $F(y,z)[x]/(x^4+y^4+zy)$?

Why is the fraction field of $F[x,y,z]/(x^4+y^4+zy)$ equal to $F(y,z)[x]/(x^4+y^4+zy)$? Is there a general formula to compute the fraction field of $F[x_1,...,x_n]/(f)$ or $F[x_1,...,x_n]/(f_1,...,f_m)$
roi_saumon
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Why assume $R$ Noetherian for the primary decomposition of $R$-modules?

When considering primary decomposition in modules, why is it often assumed that $M$ is a finitely generated module over a Noetherian ring $R$ and not only that $M$ is Noetherian? As I understand this extra assumption is not needed to prove the…
harajm
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Hilbert Nullstellensatz, Eisenbud's proof

I am trying to understand the proof on Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. The theorem I am trying to prove is: Let $k$ be an algebraically closed field. If $I \subset k[x_{1},...x_{n}]$ is an ideal, then…
Mathstudent
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Understanding a proof: Artinian ring implies semilocal

Let $R$ be an artinian ring. Then $R$ is semilocal. $(\operatorname{rad}(R))^n=0$ for some $n$. Proof: Let $I=\text{rad}(R)$. By \hyperref[3.15]{3.15}, $\exists n$ so that we have $\operatorname{ann}_{\overline{R}} \overline{I}=\overline{0}$,…
Username Unknown
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how to show that $E(A,B_1\oplus B_2)\cong E(A,B_1)\times E(A,B_2)$

Show that $E(A,B_1\oplus B_2)\cong E(A,B_1)\times E(A,B_2)$.$E(A,B)$ here means the set of equivalence classes of extensions of A by B.It's a exercise from GTM 4 ,Chapter 3,but I don't know how to prove it.
Daniel Xu
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Suppose an ideal $q$ is $p$-primary, prove or disprove that if $(q:x)=q$, then $x$ is not in $p$

Suppose $A$ is a commutative ring with unity and an ideal $q$ of $A$ is $p$-primary, i.e. $\sqrt{q}=p$. It is known that for $x \in A $, we have if $x \not\in q$, then $(q:x)$ is $p$-primary. if $x\not\in p$, then $(q:x)=q$. I was wondering…
Bingyan Liu
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$M \bigotimes_R (\Pi_{i\in I} B_i) \ncong \Pi_{i \in I} (M \bigotimes B_i)$

let M and Bi be R-MODULE for all i in I Show that $M \bigotimes_R (\Pi_{i\in I} B_i) \ncong \Pi_{i \in I} (M \bigotimes B_i)$ Take $R=\mathbb{Z}$, $M=\mathbb{Q}$ and $B_n =\frac{\mathbb{Z}}{P^n \mathbb{Z}}$ , n > 0
Aaaaaa
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Let $I$ be an ideal in a Noetherian ring. Show that either $I$ contains an $R$-regular element or else $aI=0$ for some $0\neq a\in R$.

Let $I$ be an ideal in a Noetherian ring. Show that either $I$ contains an $R$-regular element or else $aI=0$ for some $0\neq a\in R$. How would I prove this? Also what does $aI=0$ mean?
Username Unknown
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2 "equivalent" definitions of $\mathbf{C}$-algebras.

In my class professor defined a $\mathbf{C}$-algebra (for our purposes) as being a commutative unital ring $R$, that is also a $\mathbf{C}$-vector space in a compatible way (with certain associativity and distributivity identities linking the…
trynalearn
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Maximal ideal of Dedekind domain

Let $\Lambda$ be a Dedekind domain and $\mathcal{m}$ be a maximal ideal of $\Lambda$. Is it possible that $\mathcal{m}=\mathcal{m}^2$? If not, how can I prove it?
user8779
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Is an "infinite upper unitriangular" matrix always invertible?

Let $A$ be a commutative ring, $(I, \leq)$ some partially ordered set, and $V$ a free $A$-module with basis $\{e_i\}_{i \in I}$. Suppose that there is a set of elements $\{x_i\}_{i \in I} \subseteq V$ satisfying $$ x_i = e_i + \sum_{j > i} a_{ij}…
Joppy
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An isomorphism $Hom_B(m/m^2, T)\cong Hom_A(m,T).$

Let $m$ be an ideal of a ring $A$. Set $B:=A/m$. Then, $m$ and $m/m^2$ have natural structures of $A$ and $B$ respectively. For any $B$-module T which also has a natural structure of an $A$-module, I want to prove that $$Hom_B(m/m^2, T)\cong…
An Khuong Doan
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Intersection of Dedekind Domains

If $ A $, $ B $ are subrings of a field $ K $ that are Dedekind domains, is it true that $ A \cap B $ is also a Dedekind domain? Case in point: $ \mathbb{Z}_{(3) } , \mathbb{Z}_{(5) } \subset \mathbb{Q} $ and their intersection is again a…
Hodge-Tate
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Atiyah-Macdonald 11.20 and 11.21

Here, shouldn't the $x_i$s map to $x_i + \mathfrak{q}^2$, rather than to $x_i +\mathfrak{q}$? If not, then everything would be mapped to degree $1$ elements. This is another one: Here, where is the fact that $k$ maps isomorphically onto…
Jehu314
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Lemma about transcendence basis

Let $k$ be a field of characteristic zero. Let $R=k\big[z_1,...,z_p\big]$ a $k$-algebra of finite type which is a domain. Let $F$ be its field of fractions and $x_1,...,x_n$ a transcendence basis of $F$ over $k$ which verifies: $\forall…
BlueCharlie
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