# Questions tagged [commutative-algebra]

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

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### Is $(5)$ a prime ideal of $\frac{\mathbb{Z}[X]}{X^5 +2X +2}$?

Let $\alpha$ be an algebraic integer which satisfies the polynomial $X^5 + 2x + 2=0$. Then we have $\mathbb{Z}[\alpha]$ is isomorphic to $\frac{\mathbb{Z}[X]}{X^5 +2X +2}$. I then want to work out ...
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### Division in $A[[x]]$

I was looking for a division in a ring of formal power series. Specifically, let be $A$ a commutative ring with unit. Take $A[[x]]$ and $f\in A[x]$ a monic polynomial not invertible in $A[[x]]$ is so ...
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### How to find the colon of two (monomial) ideals

This question was asked in my commutative algebra assignment and I need help in solving it. Let $A= \mathbb{F}[x,y]$ , $I=(x^2y ,xy^2)$, $S=(y^2)$. Then what is $I:S$? So, I need to find elements (...
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### The extension of an invertible ideal to the ring of fractions is an invertible ideal

I am working on the following exercise: Let $I$ be an invertible ideal of a domain $R$, $S \subseteq R$ a multiplicative subset of $R$. Then $I_S$ is an invertible ideal of $R_S$ . My attempt looks ...
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### Are finite sets of points in projective space non-singular.

Let's say we have a projective algebraic set $X = \{p_1,...,p_n\}$ that's just a finite set of points. Is X non-singular? My understanding is that a variety/algebraic set is non-singular if the ...
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### Algebraic proof of pure transcendence and unirational varieties

Using the following definition: A variety is said to be unirational whenever there is a dominant rational map $\phi: \mathbb{P}^n\dashrightarrow X$ I was trying to prove that $X$ being uniration is ...
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### Is the homogenous ideal of a complete intersection always radical?

Let's say we have a complete intersection Z in $\mathbb{P}^2$ defined by two curves $f_1,f_2$ that meet transversely in the maximum number of points allowed by Bezout's theorem. Is the ideal (f_1,f_2) ...
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### If $f(x) = g(x)h(x)$ and $f(x)$ has non-negative valuation then $g(x)$ and $h(x)$ also have non-negative valuation

I am stuck at the following exercise: Let $v$ be a valuation on $K$ and $f, g, h$ monic polynomials in $K[x]$ with $f(x) = g(x)h(x)$. If $f \in R_v[x]$ then $g$ and $h$ are in $R_v[x]$. My attempt ...
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### Ideal generated by regular sequence is radical?

Let $I=(f_1,...,f_n)$ be an ideal generated by a regular sequence in $k[x_0,...,x_n]$. Then $I=\mathrm{rad}(I)$? I am reasonably confident this is true but I've been having a lot of trouble coming up ...
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### $\Bbb{Q}/\Bbb{Z}$ as a $\Bbb{Z}$-module has no free non-trivial submodules

Show that $\Bbb{Q}/\Bbb{Z}$ as a $\Bbb{Z}$-module has no free non-trivial submodules. I already proved that $\Bbb{Q}/\Bbb{Z}$ is not finitely generated as a $\Bbb{Z}$ module. I want to prove the ...
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### What are generators of a module?

Are those the basis of a module (so that every element of a module is a linear combination of those elements aka "generators")? Or are those the elements from the set from which our given ...
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### What does it mean (given that $M$ is an R-module), that $M=0$? That it is a module generated by the zero of ring $R$?

I've seen this kind of notation in the context of Nakayama's Lemma, but I don't get it if this means that M is a module generated by R's zero, or if this means that M is simply the empty set?
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### Short exact sequences. [closed]

Let R be a commutative ring with unity and I be an ideal in R. If x is an element of R not lying in I, then how to prove the following sequence is an exact sequence. Here $\psi(\bar{r})=x\bar{r}$ and ...
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### Zero dimensional quotient of a local Noetherian ring

Let $(R,m)$ be a local Noetherian integral domain. Let $I$ be an ideal in $R$, so $I \subseteq m$. Then $R/I$ is also a local Noetherian ring, see (and the image of any surjective ring homomorphism of ...
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### Misunderstanding of an exercise in Görtz-Wedhorn: When do the nilradical and Jacobson radical coincide?

Exercise 2.3 of Görtz-Wedhorn, Algebraic Geometry I, states that the nilradical of $A$ is equal to the Jacobson of $A$ if and only if every non-empty open subset of $\operatorname{Spec}A$ contains a ...
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### If the vector space dimension of $\mathbb{C}[[x,y]]/I$ over $\mathbb{C}$ is finite, then $I$ contains power of $(X,Y)$

I am trying to understand the proof which goes like this. If $\mathrm{dim}_{\mathbb{C}} \ \mathbb{C}[[x,y]]/I$ is finite, then $\mathbb{C}[[x,y]]/I$ has a finite composition series whose ...
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### Why the image of any $A$-regular sequence under $f$ is a $B$-regular sequence, $f: A \to B$ is flat.

The second answer to this question claims: "If $f: A \rightarrow B$ is flat, then obviously the image of any $A$-regular sequence under $f$ is a $B$-regular sequence. This can be seen by ...
Let $k$ be a field and let $R$ be a UFD, which is a $k$-algebra. Let $w$ be an algebraic element over $R$, namely, there exists a polynomial $f(T) \in R[T]$ such that $f(w)=0$. Denote $S=R[w]$. ...