Questions tagged [commutative-algebra]

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

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1answer
41 views

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 ...
2
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1answer
48 views

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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1answer
28 views

A short exact sequence with an ideal

Let $R$ be a UFD and $I=\langle a,b\rangle\subseteq R$ an ideal, where $a,b\in R$ s.t $\operatorname{gcd}(a,b)=1$. I want to find a short exact sequence $0\rightarrow R\rightarrow R^2\rightarrow I\...
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0answers
41 views

Quotients of multivariate polynomial rings - is $k[x][y]/(y-x^2) \cong k[y][x]/(y-x^2)$?

This question is motivated by exercise 1.1 in Hartshorne Algebraic Geometry. One has to prove that $k[x,y]/(y-x^2)$ is isomorphic to a polynomial ring in one variable. I know that this can be done by ...
2
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1answer
34 views

Understanding Proposition 1.1 Chapter XII in Lang Algebra

$\textbf{Proposition 1.1}$. Let $|\ |_1$ and $|\ |_2$ be non-trivial absolute values on a field $K$. They are dependent if and only if the relation $|x|_1<1$ implies $|x|_2<1$. If they are ...
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1answer
18 views

Some true or false questions regarding maximal ideals and polynomial divisibility

I saw the following question on a question paper: decide whether the following statements are true or false: (1) The ring $\mathbb{C}[X]/(X^4-9X^3+18X^2-13X+3)$ has $4$ maximal ideals. (2) In $\mathbb{...
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Unable to understand a step in an example related to tensor product

This question was told in class notes of our course in commutative algebra and I was unable to comprehend reasoning behind a step. so, I am asking it here. Consider $0 \to \mathbb{Z} \otimes \mathbb{...
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1answer
79 views

Prove $\phi$ to be isomorphism (an exercise in commutative algebra)

The question is from my course exercise of commutative algebra and I am asking here as I was unable to make any significant progress in it. $\DeclareMathOperator{\Hom}{Hom}$ Suppose there exists $f \...
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1answer
45 views

Show that $\mathbb Z[t^4,t^7]$ is not integrally closed in $\mathbb Z[t]$

To show $\mathbb Z[t^4,t^7]$ is not integrally closed in $\mathbb Z[t]$ is it fine to say that $t$ is a root of $x^4-t^4$ which is a monic polynomial with coefficients in $\mathbb Z[t^4,t^7]$, but $t\...
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Ideal polynomial equations in Macaulay2

Given a known ring $R$ and an ideal $I\subseteq R$, I would like to be able to solve an "ideal polynomial equation" of the form $$\sum_i I_i^{\alpha_i}=I,$$ i.e to find ideals $I_i$ so that ...
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1answer
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section 25 (Derivations and differentials) of Commutative ring theory by Matsumura [closed]

Matsumura Why can I make the kernel of a map of rings into a module over the image in this case?(the answer is below by Serge) This is on page 191 of Commutative ring theory by Matsumura, I don't ...
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1answer
55 views

An element in a ring that is both a unit and a zero-divisor - can you find my mistake?

I'm getting a contradiction with mathematics. Can you find my mistake? So, let $A = \mathbb{C}[x,y]/(xy)$. Then $\mathfrak{p}=(x-1,y)$ is a prime ideal in $A$, because it's a prime ideal in $\mathbb{C}...
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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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1answer
36 views

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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2answers
73 views

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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0answers
41 views

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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1answer
49 views

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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1answer
40 views

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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1answer
52 views

Let $M$ be a commutative monoid with the cancelation law. Show that an lcm doesn´t exist under these conditions.

Let $M$ be a commutative monoid with the cancelation law and suppose that $a \nsim b, x \nsim y, ax = by, ay = bx$, and $a$ and $b$ are irreducible. A first question was to show that $\gcd(ax,bx) = \...
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Is every element of $Quot(R)$ the root of a monic polynomial in $R[X]$?

In the lecture I encountered the following remark that I do not understand: Let $R$ be an integral domain and let $K = Quot(R)$. Now let $a/b \in K$ for $a,b \in R$ with $b \ne 0$. We may assume that ...
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A question on the subscheme of a $k$-scheme of finite type

I am reading Wedhorn's Algebraic Geometry. On the page 88, it says: Example 3.45. If $k$ is a field, and $X$ is a $k$ -scheme of finite type, then all subschemes of $X$ are of finite type over $k$. ...
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1answer
32 views

Definition of saturation of multiplicative set in Bosch

Let $A$ be a commutative ring with $1$ and let $S$ be a multiplicative subset of $A$. In Bosch's Algebraic Geometry book, he defines the saturation of $S$ to be the set $S'$ of "all elements in $...
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1answer
30 views

$\Bbb{Q}/\Bbb{Z}$ 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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1answer
29 views

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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35 views

A question about intersection of some ideals [closed]

‎Let ‎‎$‎R‎$ ‎be a‎ ‎Noetherian ‎ring ‎and‎ ‎‎$‎I‎$ ‎and ‎‎$‎J‎$ ‎be ‎two proper ‎ideals ‎of ‎‎$‎R‎$‎. ‎‎ Let ‎$$A= ‎\{‎\textbf{b}: \textbf{b} \text{ is an ideal of $R$ and } \exists‎‎ n \in \Bbb{N} ; ...
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1answer
20 views

Torsion-free Cohen-Macaulay modules

We say an $R$-module $M$ over integral domain $R$ is a torsion-free module if zero is the only element annihilated by some non-zero element of the ring $R$. Let $R=K[[x_1,\dots ,x_d]]$, $d>1$, be ...
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1answer
51 views

Understanding the concept of $k$-algebra [closed]

If $k$ is a field then 'a finite $k$-algebra' (i.e. finitely generated as a $k$-module) is equivalent to saying that '$k$-algebra which is also a finite dimensional vector space over $k$'?
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For a projective scheme over a base, when do cones and fibers commute?

Definitions: Let $R$ be a ring and suppose $X\subset \Bbb P^n_R$ is a closed subscheme cut out by the ideal sheaf $\mathcal{I}$ on $\Bbb P^n_R$. Then the largest homogeneous ideal of $X$ in $R[x_0,\...
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1answer
43 views

Koszul complex as a graded free resolution

I'm trying to compute the Hilbert function of a complete intersection by using the Koszul complex, but I think I'm approaching it incorrectly. If we let $R=k[x,y,z]$ and $$A= R/(f_1,f_2)$$ the Koszul ...
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0answers
38 views

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 ...
0
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1answer
23 views

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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1answer
85 views

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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36 views

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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24 views

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 ...
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1answer
52 views

When a simple algebraic ring extension of a UFD is a UFD

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 $S=R[w]$. ...
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1answer
95 views

Show that $A\simeq k[x_1,…,x_d]/I$

Let $k$ be a field and $A$ be an Artin local $k$-algebra such that $k\simeq A/M$. Then one fact is that $M/M^2$ is a finite dimensional $k$-vector space. I've saw that if $A =k[x]/(x^2)$ then $\dim_{...
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0answers
41 views

How do I compute the length of this module?

Let $k$ be an algebraically closed field. Let $A = k[x,y,z,w]$, $\mathfrak p = (x, w)$, $\mathfrak a = (x^d, w)$, for some $d \in \mathbb N$. How do I compute $(A/\mathfrak a)_\mathfrak p$? How do I ...
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2answers
102 views

Show that $\mathfrak{m}/\mathfrak{m}^2$ is a finite dimensional vector space

Let $A$ be a noetherian local ring and $\mathfrak{m}\subset A$ be its unique maximal ideal. Then $\mathfrak{m}/\mathfrak{m}^2$ is a $A/\mathfrak{m}$ vector space. I want to show such space is finite ...
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1answer
47 views

Minimal prime ideals of Cohen-Macaulay modules of positive dimension are minimal primes of the ring?

Let $(R, \mathfrak m)$ be a local ring of dimension $d>0$. Let $M$ be a finitely generated Cohen-Macaulay $R$-module (i.e., $\operatorname{depth}M=\dim M$). Then each localisation of $M$ is also ...
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3answers
68 views

$\mathbb Q$ is not a finitely generated $\mathbb Z$-module [closed]

Is it valid to say that $\mathbb Q$ is not a finitely generated $\mathbb Z$-module because $\mathbb Q$ is not finitely generated since $\mathbb Q$ is not Noetherian?
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1answer
31 views

On a Krull-intersection type problem for certain two generated ideals in local rings

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $x,y\in \mathfrak m$ such that $y$ is not a zero-divisor on $R$. Then, is it true that $\cap_{n=1}^\infty (x,y^n) \subseteq (x)$ ? By Krull ...
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1answer
41 views

Torsion-free modules over formal power series rings

Let $R=K[[x,y,z]]$ be the formal power series ring over a field $K$. We set $M=K[[x,y]]$. Then $M$ has a structure of $R$-module by $f:R\longrightarrow M $ via $g(x,y,z)\rightsquigarrow g(x,y,0)$. In ...
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2answers
66 views

Does $R \times S \cong R' \times S$ imply $R \cong R '$ for finite rings?

Let $R,R',S$ be finite (unital, associative) rings. Assume that $R \times S \cong R' \times S$ as rings. Does it follow that $R \cong R'$ as rings ? What if we assume $R,R',S$ to be commutative as ...
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0answers
27 views

Atiyah-Macdonald Exercise 1.6 generalization

I solved the following exercise from Introduction to Commutative Algebra (A&M): A commutative ring $A$ is such that every ideal not contained in the nilradical contains a nonzero idempotent. Prove ...
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0answers
35 views

Calculating the Hilbert Function of a Complete Intersection

I've hear that the Hilbert function is easy to calculate for complete intersections, but I haven't been able to find material that adequately explains why that is. More specifically, any information ...
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0answers
66 views

Do strict henselizations satisfy Going-Up?

Let $(A,\mathfrak{m})$ be a local ring. (I'd be ok assuming it's noetherian, if that's helpful, although I'd also be surprised if the answer really depended on this.) Let $A^{sh}$ be a strict ...
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1answer
38 views

Concerning the definition of a valuation map [duplicate]

In lecture I learned the following definition of valuation: Let $(K,+,\cdot)$ be a field and let $(G,+)$ be a totally ordered group. A map $v: K \longrightarrow G\cup\{\infty\}$ is a valuation if the ...
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0answers
37 views

Mod $p$ irreducibility test proof - yet another question

This is the statement of the mod $p$ irreducibility theorem: Let $p$ be a prime and suppose that $f(x) \in \Bbb Z[x]$ with $\deg f(x) \ge 1$. Let $\bar f(x)$ be the polynomial in $\Bbb Z_p[x]$ ...

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