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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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Ring of integers

I know that $\mathbb{Z} [\sqrt{3}, \sqrt{7} ]$ is not the ring of integers in $\mathbb{Q} [\sqrt{3}, \sqrt{7} ]$. But, I don't know how to explain. Can someone help in this. Thnx
user81468
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Support of a module and ideals

Let $R$ be a Noetherian ring , $M$ a finitely generated $R$-module and let $J$ be an ideal such that $Supp(M) \subset V(J)$ where $V(J) = \{P \in Spec(R) : P \supseteq J\}$. How to show there exists some $k>0$ such that $J^{k}M=0$? I know that when…
user6495
  • 3,957
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finite codimensional subalgebra of $\mathbb C[z]$

If $p_1, p_2$ are two polynomials in $\mathbb C[z]$, then when $\mathbb C[p_1, p_2]$ is finite codimensional in $\mathbb C[z]$? Are there some sufficient conditions (or NASC) on $p_1, p_2$? It is apparently clear that if $p_1 = p_2$, $\deg p_1>1$,…
belsam
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Kernel of a surjective homomorphism of free modules over a Noetherian ring

$R$ is Noetherian, $f:R^m \rightarrow R^n$ is surjective. Is $\ker(f)$ free?
lnth
  • 129
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Proposition 2.2 in Eisenbud

Can someone explain to me the proof for "This is the case iff each element $u\in U$ is a non-zero-divisor mod $J$"? I'm stuck on the forward implication. From what I understand, because we assume that $J = \varphi^{-1}(I)$ we obtain an injective…
klein4
  • 1,257
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Example of ring such that the nil radical is prime and 0 is not

I was just trying to think about an example of a ring that is not a domain and the nilradical is prime, however I could not find anyone. Thanks in advance.
user40276
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Atiyah and Macdonald Exercise 1.5(v)

If someone could check my proof for the following question, I would appreciate it! Let $A$ be a ring, and let $A[[x]]$ be the formal power series ring. Show that every prime ideal of $A$ is the contraction of a prime ideal in $A[[x]]$. Let…
klein4
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Computing length of certain module.

The length of a $R$-module $M$ is the least length of a composition series $$M=M_0\supset M_1\supset ... \supset M_n = 0,$$ where $M_j/M_{j+1}\cong R/P$ for some maximal ideal $P$ for all $j$. If $R$ is a local ring $(R,m)$, then this is equivalent…
Yuyi Zhang
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Integral and semi-local ring

Let $R$ be a ring and let $S$ be a subring of R. If $R$ is a semi-local ring and $R$ is integral over $S$, why $S$ is semi-local as well?
user6495
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Relation between different formulations of Nakayama's lemma

In Hulek's Elementary Algebraic Geometry, Nakayama's lemma is stated as follows: Let $A \neq 0$ be a finite $B$-algebra. Then for all proper ideals $m$ of $B$, we have $mA \neq A$. (Here, $A$ and $B$ are both commutative rings with identity.) My…
Cantor's Paradise
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Eisenbud Exercise 16.2 - differentials of a trivial extension of a ring by a module

Let $R$ be a ring (commutative with unit), and let $M$ be an $R$-module. Let $S = R \oplus M$ be made into a ring with the obvious product, where $M^2 = 0$. In Eisenbud's book (Commutative Algebra with a View Toward Algebraic Geometry), Exercise…
Ted
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If $A$ is Noetherian, then every fractional ideal is of the form $x^{-1} \frak{a}$ for some ideal $\frak{a}$ of $A$

[Statement] If $A$ is Noetherian, then every fractional ideal is of the form $x^{-1} \frak{a}$ for some ideal $\frak{a}$ of $A$, $x \in A$. [Attempt] I find this in Atiyah Macdonald Commutative algebra, Chapter 9 ,page 96 , Fractional…
hew
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Number of nonisomorphic $n$-dimensional artinian local $k$-algebras

This is about this question. For $n\in\mathbb{N}$ you can ask about how many nonisomorphic $n$-dimensional commutative $k$-algebras with only one prime ideal are there, where $k$ is an algebraically closed field. In that question we conclude…
LeviathanTheEsper
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Extension of homomorphism

Let $A \subset B$(integral domain), $B$ is finitely generated over $A$. Let $y_1, \cdots, y_n \in B$ algebraically independent over $A$. Then homomorphism $f:A \to \Omega$(algebraically closed field) can be extended to $A[y_1,\cdots,y_n] \to \Omega$…
Gobi
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Inequalities among heights

Let $R$ be a Noetherian ring and $I$ a non-zero ideal of $R$. Let $x\notin I$. Could someone provide me a counterexample to the following: $$\operatorname{ht}(I)\leq \operatorname{ht}(I+(x))\leq \operatorname{ht}(I)+1?$$ Here $\operatorname{ht}(I)$…
messi
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