0
votes
0answers
21 views

The radical of differential ideal

Consider a field $K$ such as $\mathbb{Q}\subseteq K$ and the ring $K[x,y]$, with the derivation $\frac{d}{dx}$. The ideal $\mathfrak a=(x^2,y^2,2)$ is a differential ideal. I have a doubt trying to ...
0
votes
3answers
40 views

Let $R$ be a PID and $I$ is a non zero proper ideal of $R$. show that if $R/I$ has no nonzero zerodivisor, then it is a field. [on hold]

Let $R$ be a PID and let $I$ be a non-zero proper ideal of $R$. Show that if $R/I$ has no non-zero zerodivisor, then it is a field.
1
vote
1answer
58 views

Need an explanation for homomorphism in commutative algebra

I'm self-learning commutative algebra following "Introduction to Commutative Algebra". When dealing with concepts like "contraction" and "extension", some exercises in this book don't specify which ...
6
votes
2answers
71 views

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,…,x_6]$ a radical ideal? Is it a prime ideal?

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,...,x_6]$ a radical ideal? Is it a prime ideal? thanks
1
vote
1answer
49 views

Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
0
votes
2answers
66 views

Checking the maximality of an ideal

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...
3
votes
0answers
62 views

is there a criterion that says whether an ideal is radical or not?

Let $R=k[x,y,z]$. Is there a criterion that says whether an ideal of $R$ is radical or not? thanks
3
votes
1answer
112 views

What are the maximal ideals of $\mathbb{Z}[t,t^{-1}]\otimes \mathbb{Q}$?

I know that $\mathbb{Z}[t,t^{-1}]$ is a localization of $\mathbb{Z}[t]$, the multiplicative set consisting of the non-negative powers of $t$. But I do not know the maximal ideals of ...
-1
votes
1answer
77 views

Number of maximal and prime ideals

Find how many prime and maximal ideals there are in the ring consisting of matrices $$M= \begin{bmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{bmatrix} $$ ...
3
votes
2answers
82 views

Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well ...
1
vote
2answers
42 views

Quotient ring is cyclic group implies every ideal is generated by 2 elements

I'm trying to solve the following exercise: Let $R$ be a commutative ring with identity. If for every ideal $\mathfrak{a} \neq 0$ of $R$ we have ($R/\mathfrak{a}$,+) is a cyclic group then ...
0
votes
2answers
75 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
2
votes
1answer
45 views

$A_{p}$ is a field when $p$ is a minimal prime and $A$ reduced

$A$ is a reduced commutative ring with unit; $p$ is a minimal prime ideal. If $S = A \setminus{p}$ , I have to show that the ring $A_{p} = S^{-1}A$ is a field. My thoughts: Since $p$ is a minimal ...
1
vote
0answers
73 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
-1
votes
1answer
66 views

Maximal multiplicative set and minimal prime ideal

Let $A$ be a ring and $P$ a prime ideal included in $A$. Show that $A \setminus P$ is a maximal multiplicative set if and only if $P$ is a minimal prime ideal of $A$. What can be the proof for this ...
0
votes
1answer
56 views

Invertible elements and maximal ideals of a localization

Let $n\in\mathbb Z$ and let $A$ be the set of integers co-prime to $n$. Denote $A^{-1}\mathbb Z$ by $\mathbb Z_{(n)}$. 1) Find the invertible elements of $\mathbb Z_{(6)}$ My attempt: let $m$ be ...
2
votes
0answers
48 views

Example of irreducible ideal which is not strongly irreducible

I have read a paper with title Ideal Theory in Commutative Semirings by Reza Ebrahimi Atani and Shahabaddin Ebrahimi Atani. In this paper we have the following definitions: An ideal I is irreducible ...
2
votes
0answers
50 views

Normal ring and unmixed ideals

Let $R$ be a commutative Gorenstein local ring , $I$ an ideal of $R$ . If $R/I$ is normal ring , then for any $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
4
votes
2answers
69 views

If $A/\mathfrak a$ is flat over $A$ then $V(\mathfrak a)$ is open. Why?

I am trying to understand the following statement. Let $A$ be a noetherian commutative ring and $\mathfrak a\subset A$ is an ideal. Suppose that the ring $A/\mathfrak a$ is flat over $A$, then ...
2
votes
1answer
63 views

Minimal primes and zero divisors

Let $R$ be a commutative local ring, $M$ a finitely generated $R$-module, and $x \in M$. Is it true that if for any $p \in$ $\operatorname{Min}(R)$ there exists $a_{p}\notin{p}$ such that $a_{p}x=0$, ...
0
votes
2answers
34 views

Spec($A$) is connected if $A$ is local

Another exercise from Balwant-Singh: Show that if $A$ is local then Spec($A$) is connected in the Zariski topology. Any hint ?
1
vote
1answer
42 views

Idempotent/Spec

I'm studying Basic Commutative Algebra by Balwant-Singh; I'm stuck on this exercise: $A$ is a commutative ring; show this $3$ conditions are equivalent: 1) $A$ contains a non-trivial idempotent 2) ...
0
votes
1answer
53 views

Is it true that an ideal is primary iff its radical is prime?

Is it true that an ideal $I$ in a commutative ring is primary iff $Rad(I)$ is prime? If not, what are some nice counterexamples?
7
votes
2answers
95 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
1
vote
1answer
71 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
2
votes
1answer
87 views

Let $I= (X_1,X_2) \cap (X_3,X_4)$. Is $ara(I)≥3$? Is $ara(I)≥4$?

This question is related to Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$ Let $R=k[X_1,X_2,X_3,X_4]$ and $I= (X_1,X_2) \cap (X_3,X_4)$. I know that ...
4
votes
1answer
35 views

If $I\leq K[X_0,\dots,X_n]$ for $K$ a field is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous?

If $I\leq K[X_0,\dots,X_n]$ (for $K$ a field, let's say algebraically closed) is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous? I'm trying to understand ...
2
votes
0answers
91 views

Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
1
vote
1answer
43 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
7
votes
1answer
115 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
2
votes
1answer
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
1
vote
1answer
19 views

Maximal element of $(I : x)$, where $x$ is in $A - I$, is prime belonging to $I$

Given that $I$ is decomposable, I am supposed to prove that any maximal element $P$ of the set {$(I : x) | x \in A - I$} must belong to $I$, i.e., $P$ is prime and for every reduced primary ...
1
vote
2answers
37 views

In $\mathbb{Z}[t]$, $Q = (4, t)$ is not a power of $M = (2, t)$

The problem of showing that Q, as above, is not a power of M, as above, rises as part of a larger problem. I'm confident about my response to the other parts, but the best justification I can come up ...
0
votes
1answer
31 views

Finitely generated ideal in boolean ring [duplicate]

A boolean ring is a commutative ring where $x^{2} = x$ for every $x$. Why in such a ring a finitely generated ideal is principal ?
4
votes
3answers
200 views

Idempotent in a local ring

Is it true that a local ring, i.e. a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
1
vote
1answer
73 views

Height unmixed ideals

Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial over field $k$ and $I$ an unmixed homogeneous ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an ...
1
vote
0answers
112 views

Primary ideals in Noetherian rings

For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$. This equals saying that ...
1
vote
1answer
32 views

Maximal (among non-principal ideals) Ideal Must be Prime

If I is an ideal which is maximal among the ones that are not principal, then I is prime. This would mean that for all $f \in R$, $(f) \subset I$. Could I then use column ideals? I was thinking ...
0
votes
1answer
30 views

Maximal Ideal Must be Prime

I am trying to prove that an ideal that is maximal with respect to not being finitely generated must be prime. What does it mean to be an ideal that is maximal with respect to not being finitely ...
0
votes
0answers
56 views

Zero dimensional ideals and their primary decomposition

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$, and $I$ a zero dimensional ideal with a primary decomposition $I=\cap Q_i$. Why is $\sum \dim_k S/Q_i = \dim_k S/I$?
2
votes
0answers
37 views

Find the projective closure of the ideal $I=\langle y-x^2,z-x^3\rangle$

When I looked at this example, my first instinct was to homogenize only the generators of $I=\langle f_1 := y-x^2,f_2:=z-x^3\rangle$ in a new variable $w$. But then, I realized that I would miss some ...
2
votes
1answer
70 views

Hilbert-Burch theorem characterizes perfect ideals of grade $2$

Bruns and Herzog in their book Cohen-Macaulay Rings, page 120 write: "The Hilbert-Burch theorem 1.4.17 identifies perfect ideals of grade $2$ as the ideals of maximal minors of certain matrices". ...
2
votes
1answer
41 views

Height of a specific maximal ideal

Let $k$ be a field, $k[x,y^2,xy,y^3]$ our ring and $\mathfrak a$ the ideal generated by $x,y^2, xy,y^3$. I want to determine the height $h(\mathfrak a)$ of $\mathfrak a$. My ideas: We see easily ...
2
votes
1answer
76 views

$\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = \{ \mathfrak{p}A_\mathfrak{p}\} $

Let $k$ be a field, $A = k[X_1,X_2,...]$, $\mathfrak{p} = (X_1,X_2,...)$, $I = (X_1^2-X_1,X_2^2-X_2,...)$, $M= A/I$. I am trying to show that $\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = ...
0
votes
0answers
52 views

Is every module a direct limit of cyclic modules?

I want to show that $M$ is $A$-flat is equivalent to $Tor_1^A(M,A/I)=0$ for every finitely generated ideal $I$. I want to show $Tor^A_1(M,N)=0$ for any $A$-module $N$. Is every module a direct ...
4
votes
1answer
50 views

Maximal ideals in the ring of eventually constant sequences of real numbers

For homework I am studying the ring $R$ of eventually constant sequences of real numbers (with multiplication and addition defined componentwise). What are the maximal ideals of $R$? By looking at ...
5
votes
1answer
65 views

Height unmixed ideal and a non-zero divisor

Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal? Background: A ...
0
votes
2answers
27 views

How does one find a minimal primary decomposition?

What exactly does it mean for a primary decomposition to be "minimal" and is the a general method to obtain such decompositions? I've tried looking at some examples but they all give very little ...
0
votes
1answer
35 views

Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
3
votes
1answer
59 views

Is there an example of commutative ring with exactly three prime ideals for which this property holds?

Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have ...