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

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2
votes
1answer
36 views

Is $K[[x]]$ an Artinian/Noetherian $K[x]$-module?

Let $K$ be a field an consider $K[[x]]$ as a $K[x]$-module. Determine if it is Artinian/Noetherian. I used the following propositions: If M is an $R$-module and $N\subseteq M$ a submodule, then ...
-1
votes
0answers
25 views

Difference between division algorithm and buchberger's algorithm

what is the main difference between division algorithm and buchberger's algorithm? I think we use the same steps where non-zero remainder is said to be the new polynomial and continue this process ...
6
votes
3answers
85 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a ...
1
vote
0answers
39 views

Better understanding regular functions on a Projective variety

Hi guys I was just looking an example from class that was left as obvious, but it is not so obvious to me. $W= V(x_1x_4-x_2x_3)= $ where $I(W)= \langle x_1x_4-x_2x_3 \rangle$ so we just picked an ...
5
votes
1answer
43 views

Is $K[x_1,\ldots,x_{n+1}]$ separable over $K[x_1,\ldots,x_n]$?

Let $R \subseteq S$ be commutative rings. $S$ is separable over $R$ if $S$ is a projective $S \otimes_R S$-module (under $\mu: S \otimes_R S \to S$ defined by $\mu(s_1 \otimes s_2)=s_1s_2$). Let ...
3
votes
2answers
41 views

$R/I \otimes_A R/I \cong (R \otimes_A R)/(I \otimes_A I)$?

Let $f: A \to R$ be a homomorphism of commutative rings, and let $I$ be an ideal of $R$. Is it true that $R/I \otimes_A R/I \cong (R \otimes_A R)/(I \otimes_A I)$ ? After obtaining the surjection ...
1
vote
1answer
35 views

Can we make sense, in general, of taking a quotient by multiple ideals?

I feel that this is a rather silly question, stemming from a fundamental misunderstanding of quotients, but I'm not quite able to make it precise. My question is: given two ideals ...
1
vote
2answers
47 views

Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$

Let $A \subseteq B \subseteq C$ be commutative rings (noetherian integral domains, if this helps). Assume $C$ is a smooth $A$-algebra. Is it true that $C$ is a smooth $B$-algebra?
2
votes
1answer
51 views

Using Koszul complex [closed]

Let $A$ be a Noetherian local ring of dimension $t$ with maximal ideal $\mathfrak{m}$. If $J\subset A$ is an $\mathfrak{m}$-primary ideal then we have the following complex for $n\in \Bbb N$: ...
1
vote
1answer
38 views

Minimal polynomial of an algebraic element over any domain

Let $A \subseteq B$ be integral domains ($A$ is not necessarily a UFD). Assume $b \in B$ is algebraic over $A$, namely: $a_nb^n+\cdots+a_1b+a_0=0$ for some $a_i \in A$ not all zero. My (trivial) ...
1
vote
1answer
23 views

Minimal prime ideal over $(x,P)$

Let $R$ be a Cohen-Macaulay ring with a prime ideal $P$ and an element $x\notin P$ such that the ideal $\langle x,P\rangle$ generated by $x$ and $P$ is not the whole of $R$. If, moreover, $Q$ is a ...
0
votes
1answer
34 views

A question about $\operatorname{Tor}_i$

Suppose P'$\to$M is a projective resolution of M. And P'$\bigotimes$C is a complex and the definition of $Tor_i$ is $h^i$(P'$\bigotimes$C). However I am confused about $Tor_i$. As tensor product is ...
0
votes
0answers
15 views

What is the additive inverse of a 2-adic Witt vector?

The ring of $p$-adic Witt Vector is defined as follows: First, let $X_0, X_1, \cdots$ be an infinite sequence of unknowns and put \begin{equation*} W_n = X_0^{p^n} + pX_1^{p^{n-1}} + \cdots + ...
1
vote
1answer
52 views

local cohomology and radical of ideal

Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ_I$ as: $\Gamma_I(M)=\bigcup_{n\in N} (0:_MI^n).$ When $R$ is Noetherian, ...
1
vote
1answer
23 views

Principal ideals containing an ideal in a Noetherian integral domain

Let $R$ be a Noetherian integral domain and $I$ a nonzero ideal consisting only of zero divisors on $R/(x)$, where $x$ is a nonzero element of $I$. Could we always find an element $y\notin (x)$ such ...
-1
votes
1answer
75 views

Exercise $1.8$ of chapter one in Hartshorne.

In exercise 1.8 of chap I in Hartshorne algebraic geometry, Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y ...
3
votes
1answer
61 views

Krull dimension of Nagata rings

I want any help concerning proof of the following theorem of Nagata: Let $S$ be the set of all $f\in R[x]$ with the property that the coefficients of $f$ generate the unit ideal. Then (a) $S$ is a ...
2
votes
0answers
35 views

When is the set $A=\{a+s|a\in I , s\in S \}$ a prime ideal of R?

Let $R$ be commutative ring with identity, $I$ an ideal of $R$, and $S$ a subset of $R$. Under what conditions is the set $A=\{a+s\mid a\in I , s\in S \}$: 1- an ideal of $R$? 2- a prime ...
8
votes
0answers
92 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
0
votes
0answers
54 views

In what conditions every ideal is an extension ideal?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
2
votes
2answers
38 views

$p$(ain)-adic number sequence

I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH! Jokes aside, I am working on this question Show ...
15
votes
0answers
157 views

Can any commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

Let $S$ be a commutative ring with identity with $\operatorname{char}S=p$, where $p$ is a prime number. I wonder if we can always find a ring $R$ such that $\operatorname{char}R=0$ and $R/(p)\cong ...
0
votes
1answer
22 views

Equivalence of definitions of dimension in integral domains.

If $R$ is a finitely generated algebra over a field $k$ that is an integral domain, it is known that the Krull dimension of $R$ is equal to the transcendence degree of the field of fractions of $R$ ...
2
votes
1answer
48 views

Equivalence of line bundles and $\mathbb{G}_m$-torsors

This appears to be a duplicate of (half of) this question, but it received no attention so I'll try again. Given a line bundle $L\to X$ on a scheme $X$ over a field $k$, I am to show that ...
0
votes
0answers
56 views

What is the geometric interpretation of a P-primary component of an affine k algebra?

Let $R= K[x_1, x_2,...,x_n]$ for some algebraically closed field K. If $I \subset R$ is an ideal, and P is a prime minimal over I, I know that $Z(P)$ is a maximal irreducible subset of $Z(I)$. But ...
5
votes
1answer
60 views

Separability of $A \subseteq C$ implies separability of $B \subseteq C$, where $A \subseteq B \subseteq C$

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$). My ...
7
votes
2answers
244 views

Category of binomial rings

A binomial ring is a commutative ring $R$ such that (1) the additive group of $R$ is torsionfree and (2) $n!$ divides $x(x-1)\dotsc(x-n+1)$ for all $n \in \mathbb{N}$ and $x \in R$. We may then define ...
3
votes
0answers
42 views

Properties of polynomials that are polynomial conditions on the coefficients

There are many occasions where we can check whether a (set of) polynomial(s) $f_i$ satisfies certain properties, simply by evaluating a fixed polynomial on the coefficients of the $f_i$. Many times, ...
3
votes
0answers
42 views

Kähler differentials, define valuation?

See my previous question for a definition of the $K$-module of Kähler differential $\Omega_{K/k}$. This question is sort of a follow up on it. Suppose $k$ is a field of characteristic $0$, $R$ is a ...
2
votes
0answers
58 views

Kähler differentials, define valuation?

See here for a definition of the $R$-module of Kähler differential $\Omega_{R/k}$. Suppose $k$ is a field of characteristic $0$, $R$ is a $k$-algebra, and let $K$ be a finite extension of $k(x)$. If ...
0
votes
0answers
18 views

Result showing that a certain valuation ring in some function field has to be a DVR?

I know that if $R$ is a valuation ring such that $0 \to \mathbb{C} \to R$ is a left-split exact sequence, then there exists a discrete valuation ring $C$ with $R \subset C$ so that $0 \to \mathbb{C} ...
2
votes
1answer
41 views

Finding roots of an irreducible polynomial in a ring that is not a domain.

Let $A$ be a commutative ring, and $p\in A[X]$ a polynomial of degree $d>0$. If $A$ is an integral domain, we can find a ring $B$ such that $A\subseteq B$ and $p$ has a root in $B$. For example ...
2
votes
1answer
37 views

Isomorphic function fields of projective curves, bijection of points.

Suppose curves $C$, $D \in \mathbb{CP}^2$ are nonsingular. If their function fields are isomorphic, i.e. $K_C \cong K_D$, then do we necessarily have a bijection of points on $C$ and $D$? Can we do ...
3
votes
2answers
100 views

Show that $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. [duplicate]

I am trying to prove that the ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. I have an hint, that suggests to find an isomorphism between $\mathbb{C}[x,y]/(x^2+y^2-1)$ and $\mathbb{C}[e^{it},e^{-it}]$, ...
3
votes
3answers
107 views

Exercise 1.9 in Hartshorne - is my initial attempt a good start?

Hartshorne's Chapter 1, exercise 1.9 asks us to show that irred. components of $Z(\mathfrak a)$ have dimension $\geq n-r$ if $\mathfrak a$ is an ideal generated by $r$ elements. I think I've reduced ...
0
votes
1answer
24 views

Factorization of a ring morphism for artinian rings

Let $(R,m)$ be a complete local $k$-algebra. Let $A$ be a local artinian algebra with residue field $k$. Then, since $A$ is artinian, any map $f:R \rightarrow A$ factorizes as $R \rightarrow R/m^n ...
3
votes
1answer
41 views

Commutative Hereditary Rings

Is it true that the ring $\mathbb Z/n\mathbb Z$ ($n≠0$) is hereditary if and only if $n$ is square-free? The "if" part is OK to me because any field $\mathbb Z/p \mathbb Z$ ($p$ prime) is a PID, ...
4
votes
0answers
40 views

Definitions of the ring $R/(x_0^\infty,\ldots,x_{n-1}^\infty)$

Let $R$ be a ring, and let $I = (x_0,\ldots,x_{n-1})$ be a finitely-generated ideal inside of $R$, generated by a regular sequence. In algebraic topology one often encounters a ring, usually denoted ...
0
votes
0answers
47 views

Is an irreducible ideal in $R$ irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
2
votes
1answer
27 views

$A\subseteq B$, $B$ integral over $A$, $\mathfrak{q}_{1}\subsetneq\mathfrak{q}_{2}$, then $A\cap\mathfrak{q}_{1}\subsetneq A\cap\mathfrak{q}_{2}$.

Let $A, B$ be commutative rings such that $A\subseteq B$ and $B$ is integral over $A$. I want to prove that if $\mathfrak{q}_{1},\mathfrak{q}_{2}$ are prime ideals of $B$ such that ...
0
votes
1answer
31 views

Another question about proposition 5.15 of Atiyah-MacDonald.

I have a trouble in understanding the proposition 5.15 in the book from Atiyah and MacDonald. I see that some time ago another user asked a similar question (Proposition 5.15 Atiyah Macdonald: ...
2
votes
1answer
31 views

If $\overline{k}$ is an algebraic closure of a field $k$, then $\overline{k}[x_{1}, \dots, x_{n}]$ is integral over $k[x_{1}, \dots, x_{n}]$.

I want to prove that if $\overline{k}$ is an algebraic closure of a field $k$, then $\overline{k}[x_{1}, \dots, x_{n}]$ is integral over $k[x_{1}, \dots, x_{n}]$. It is used in exercise 11.3 of the ...
2
votes
0answers
37 views

Characterization of tensor products of fields

For which commutative rings $R$ are there field homomorphisms $L \leftarrow K \to L'$ (not assumed to be algebraic or anything) such that $R \cong L \otimes_K L'$? Is there an intrinsic ...
2
votes
0answers
37 views

Subring of all elements represented by quotients of function field.

Suppose $K_C$ is the function field of a curve $C$ and $p \in C$. Let $\mathcal{O}_k \subset K_C$ be the subring of all elements represented by quotients $G/H$ where $G, H \in \mathbb{C}[x, y, z]$ are ...
2
votes
1answer
33 views

Natural isomorphism between curve and its projective completion?

If $C \subset \mathbb{C}^2$ is an irreducible affine curve and $\hat{C} \subset \mathbb{P}_2$ is its projective completion, is there necessarily a natural isomorphism of function fields $K_C \cong ...
3
votes
2answers
112 views

Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?

Let $R$ be a commutative integral domain, $I,J,K$ three ideals of $R$ with $I\neq (0)$ being finitely generated. Then does $IJ=IK$ imply $J=K$? With Nakayama lemma, I can prove it if one of $J$ and ...
3
votes
1answer
47 views

How can affine coordinate rings be canonically identified as $k$-algebras?

Exercise 1.5 of Hartshorne asks us to show (in one direction) that any affine coordinate ring $k[x_1,\dots,x_n]/I(Y)$ is a finitely-generated $k$-algebra with no nilpotents. The second part is quite ...
0
votes
0answers
32 views

Smooth morphism and completion of DVR

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field and $\hat{R}$ its $m$-completion, where $m$ is the maximal ideal. Is it true that the induced morphism ...
1
vote
1answer
42 views

Exact sequence of modules and taking the quotient

Let $A$ be a commutative ring and $\text{Spec}\,A=\bigcup\limits_{i=1}^mD(f_i)$ be a covering by principal open sets. Show that the sequence of modules $$M\stackrel{\alpha}\to ...
0
votes
0answers
32 views

DVR and its fraction field

Let $k$ be a complete discrete valuation field with algebraically closed residue field. We know that its maximal unramified extension $k^{\mathrm{unr}}$ need not be complete. But can the ring of ...