Questions tagged [maximal-and-prime-ideals]
For questions about prime ideals and maximal ideals in rings.
1,783
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length of $\mathbb{C}[x]/(x^{50}+x+1)\mathbb{C}[x]$ as an $\mathbb{C}[x]$-module
I was wondering how to compute the length of $\mathbb{C}[x]/(x^{50}+x+1)\mathbb{C}[x]$ as an $\mathbb{C}[x]$-module or how to proceed in general when you have such a large quotient, as you can´t just ...
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Compute $\dim k[X,Y,Z]/(XYZ,Y^2)$
Compute $\dim k[X,Y,Z]/(XYZ,Y^2)$
I had to solve this in my exam on commutative algebra, but without a proof. Is there an easy way to compute this?
I need a maximal chain of prime ideals. The prime ...
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1
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Commutative algebra: Irreducible components of $\operatorname{Spec}(A)$
Let $k$ be a field of characteristic other than $2$.
Consider the ring $$A:= k[X,Y]/(X(Y+1),X(Y+X^2)).$$
Describe all the irreducible components of $\operatorname{Spec}(A)$.
It does not look that ...
- 33
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Inverse image of maximal ideals under finite type ring maps.
All rings are commutative with $1$.
I am trying to find a name of such ring $R$ with the following property:
For any finite type ring map $f: R\to A$, inverse image
$f^{-1}(\mathfrak m)$ of any ...
- 1,966
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If $A\subset B$, $\mathfrak{p}$ be a prime of $A$, $x+\mathfrak{p}B$ is a unit, irreducible $f\in A[X]$ has $f(x)=0$, must $[X^0]f\notin\mathfrak{p}$?
Let $A$ be a commutative ring with unity, $B$ be an integral extension of $A$ (that is, every element in $B$ is a root of a monic polynomial in $A[X]$). Let $\mathfrak{p}$ be a prime ideal of $A$. ...
- 795
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"Simpler" proof of general Nullstellensatz for Jacobson rings.
Assume all rings/algebras are commutative with identity. "$\hookrightarrow$" means injection and "$\twoheadrightarrow$" means surjection. $A_f$ is the localization of $A$ on the ...
- 1,966
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18
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Homogeneous prime ideals in $(S_\star)_f$ map bijectively to homogeneous prime ideals in $(S_\star)$ not containing $(1,f,f^2,...)$?
As the title suggests, why do homogeneous prime ideals in $(S_{\star})_{f}$ map bijectively to homogeneous prime ideals in $S_{\star}$ not containing $(1,f,f^2,...)$?
What I tried: I know that prime ...
- 429
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If every maximal ideal of $R$ have the same height, then does the same property hold for $R/P$ for every minimal prime $P$ of $R$?
Let $R$ be a commutative Noetherian ring such that every maximal ideal of $R$ has the same height. This also implies that $\dim R$ is finite and equals the common number of the height of the maximal ...
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Maximal ideals of ring of continous functions on $\mathbb{R}$
Can we characterize all the maximal ideals of the ring of real valued continuous functions on $\mathbb{R}$? One type of such ideal is the evaluation at a certain point and another is the maximal ideal ...
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Maximal homogeneous ideals in graded ring A[T]
Let $A$ be a commutative algebra with unit. Let $A[x]$ be the polynomial algebra with coefficients in $A$ with the standard gradation (by degrees).
I have the following questions.
What are maximal ...
- 209
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Characterization of closed points $x$ of affine $k$-varieties with $[\kappa(x):k]=1$
$\def\frm{\mathfrak{m}}
$Let $k$ be a field. Let $\frm\subset k[x_1,\dots,x_n]$ be a maximal ideal of the polynomial ring. We get a field extension
$$
\label{eq}\tag{1}
k\hookrightarrow k[x_1,\dots,...
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Ideals in a Von-Neumann regular ring
I want to show the following result directly:
Let R be a regular ring then every ideal is the intersection of
maximal ideals containing it.
I know that every ideal of a regular ring is a z-ideal (...
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1
answer
34
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Maximal ideal of $C(\mathbb{R})$
Is the ideal $I$ defined as follows a maximal ideal in $C(\mathbb{R})$?
$$I=\{f\in C(\mathbb{R}):\exists N\in {R} , f(x)=0 \forall x>N\}$$.
Where $C(\mathbb{R})$ is ring of all real valued ...
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Give an example about associated primes where two containments are proper.
I want to ask a question about associated primes. This question is just for my curiosity.
Let $R$ be a commutative ring with $1$. Let $M$ be a $R$-module. We use $\mathrm{Ass}_R(M)$ to denote the ...
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Prime Ideals of Products of $\mathbb Z_p$
I know that the Prime Ideals of $\mathbb Z_p\times\mathbb Z_p$ are $\langle(1,0)\rangle $ and $\langle(0,1)\rangle$, but what would it be for $\mathbb Z_p\times\mathbb Z_p\times\mathbb Z_p$?
Would it ...
- 1
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Commutative ring without unit and maximal ideals
I'm stuck with the following:
Let $R=\{x\frac{f(x)}{g(x)}\,:\,f(x),g(x)\in\mathbb{R}(x),\,g(0)\neq 0\}$
Show that $R$ has not maximal ideals.
My principal problem is that $R$ has not unital, so if $M$ ...
- 7,477
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Stone's criterion for distributive lattices
A lattice $L$ is distributive iff for every $a\ne b$ in $L$ there is a prime ideal $P\in\textrm{Spec}(L)$ s.t. either $a\in P\not\ni b$ or $b\in P\not\ni a.$
Right to left. Since $D:L\to2^{\textrm{...
- 1,273
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Radicals ideals equal in field extension
If $f,g \in F[x]$ ($F$ is a field) are two polynomials, then we can obviously have $\text{rad} (f)=\text{rad} (g),$ for instance, $f=x$ and $g=x^2.$ However if the polynomials are irreducible, then ...
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Prove of specific prime ideals between a strict inclusion of prime ideals
Statement. Let $R$ be a commutative ring with prime ideals $\mathfrak{p}$ and $\mathfrak{q}$. If $\mathfrak{p}\subsetneq\mathfrak{q}$, then there are prime ideals $\mathfrak{p}_1$ and $\mathfrak{q}_1$ ...
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Chebotarev's Density Theorem, Equidistribution of Prime Ideals, and Class-Field Theory
I am working on a senior thesis, and my advisor told me to look into the theory that prime ideals in a number field of norm less than $N$ are evenly distributed across ideal classes.
I've looked at ...
- 389
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Dimension of $(A[[X]])_\mathfrak{m} \geq k+1$, regular ring, chain of prime ideals
Assume $A$ is a regular ring and $m$ a maximal ideal of $A$. We define $R= A[[X]]$.Then $\mathfrak{M}=mR + XR$ is a maximal ideal of $R$. Lets assume $h(mA_m) = k$.
I want to show, that
\begin{align*}
...
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1
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35
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Localization in power series
Let $A$ be a comm. ring with unity.
Say $\mathfrak{M}$ is a maximal ideal of $A[[X]]$.
Is following statement generally true?
\begin{align*}
(A[[X]])_\mathfrak{M} \cong (A_{\mathfrak{M} \cap A}[[X]])_\...
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answer
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Understanding a lemma about minimal prime ideals (from McCoy's Rings and Ideals).
I am reading N. H. McCoy's book Rings and Ideals. In the section on minimal prime ideals belonging to (containing) an ideal, he proves the following lemma (lemma 3 in the section):
THEOREM A set $\...
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Independence complex of a prime ideal is a matroid
Let $k$ be a field and $I \subseteq k[x_1, \ldots, x_n]$ be an ideal.
Definition. A subset $ \underline{u} \subseteq \{ x_1, \ldots, x_n\} = \underline{x} $ of variables is independent modulo $I$ if $...
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Let $R$ be an integral domain. Let $I$ and $J$ be nonzero prime ideals such that $I\subsetneq J$. If $I$ is principal, show that $J$ is not principal. [duplicate]
I'm a little stuck on this question from my Abstract Algebra homework and I would like a hint (read: please don't solve this for me, I just want a little hint to get me on track). Here is the question ...
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Justify the use of Zorn's Lemma in a proof
I was reading this proof of how the nilradical of a ring is the intersection of all prime ideals of the ring.
https://artofproblemsolving.com/wiki/index.php?title=Nilradical
In the proof, it says &...
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Show that an ideal $M \ne R$ in a commutative ring $R$ with unity is maximal if and only if for every $r∈R−M$, there exists $x∈R$ such that $1_R−rx∈M$
Show that an ideal $M \ne R$ in a commutative ring $R$ with unity is maximal if and only if for every $r∈R−M$, there exists $x∈R$ such that $1_R−rx∈M$
Proof : Sppose that for every $r ∈ R - M$, there ...
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Which of the following statements are correct regarding $I(x)?$
Let $X=(0,1)$ be the open unit interval and $C(X,\mathbb R)$ be the
ring of continuous functions from $X$ to $\mathbb R.$For any $x\in
(0,1),$let $I(x)=\{f\in C(X,\mathbb R):f(x)=0\}.$Then which of ...
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Properties of an ideal located between two prime ideals.
Let $R$ be a commutative ring with unity. Let $P,I,Q$ be ideals of $R$ such that $P$ and $Q$ are prime ideals and $P\subseteq I \subseteq Q$. Let $a\in R$ and $b\in R\setminus Q$. I would like to have ...
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Indecomposable injective modules
Let $R$ be a ring with $1$. It's well known that if $M$ an indecomposable injective right $R$-module, then $M\cong E(R/\mathfrak{p})$ for some prime ideal $p\subset R$ where $E(R/\mathfrak{p})$ is the ...
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On prime ideals in $\mathbb{F}_{11}[X]$
Consider the ideals $I=(X^4+4X^3+6,X-1)$ and $J=(X^4+4X^3+6,3)$ in $\mathbb{F}_{11}[X]=(\mathbb{Z}/11\mathbb{Z})[X]$.
Are $I$ and $J$ prime ideals in $\mathbb{F}_{11}[X]$?
Attempt: For the ideal $I$ ...
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Are square of a maximal ideal and a maximal ideal coprime?
Let $R$ be a commutative ring with $1$. Let $I, J \subset R$ be different maximal ideals. Is it true that $I^2$ and $J$ are coprime?
It is well known that $I$ and $J$ are coprime seen by the fact $I + ...
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Is $(X^4+4X^3+6,X-1)$ a prime/maximal ideal in $\mathbb{Z}[X]$ [duplicate]
As the title states, I'm curious if the ideal $I=(X^4+4X^3+6,X-1)$ is prime/maximal in $\mathbb{Z}[X]$.
I think that this ideal is not prime/maximal in $\mathbb{Z}[X]$, because the prime ideals in $\...
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Let $I$ be a non null ideal in $A$ P.I.D, if $A/I$ is an integer domain it is also a field? [duplicate]
I have no work to show. This fact is presented in my book as an obvious corollary of the following theorem:
Theorem:
If I=$(p)$ is an non null ideal in A P.I.D, $I$ is prime iff it is maximal.
Thanks ...
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Serre's condition $(S_n)$ locus is stable under generalization?
Let $R$ be a Noetherian ring and $n\ge 1$ an integer. Consider the set $S_n^R:=\left\{\mathfrak p \in \text{Spec}(R) \text{ }| \text{ } \text{depth } R_{\mathfrak p} \ge \inf \{n, \dim R_{\mathfrak p}\...
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For every positive integer n, show that there is a ring with exactly n ideals. [closed]
Please may I have a hint for this question? I am trying this by induction at the moment, with the base case being {0}. I have assumed the result holds for integers up to and including n-1. My thought ...
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Complex conjugate of prime element generates the same prime ideal? [closed]
Let $R_K$ be ring of integers of imaginary quadratic field $K$.
Let $P$ be a prime ideal of $R_K$.
Suppose $ \pi$ be prime element of $R_K$ which generates $P$.
Let $ \pi'$ be complex conjugate of $ \...
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Find all maximal ideals of a ring.
I'm trying to find all maximal ideals of a ring
$R=\left\{\left(\begin{array}{cccc}
a & 0\\
b & a\\\end{array}\right),\ a,b \in \mathbb{Z}_{10} \right\} $.
I assume that matrix unit are used ...
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On maximal ideals in $\mathbb{Q}[X,Y]$ and $\mathbb{C}[X,Y]$
Show for the ideal $I=(y-x^2,xy+10x-5)$ if it's maximal in $\mathbb{Q}[X,Y]$ or in $\mathbb{C}[X,Y]$.
To show that $I$ is a maximal ideal I would usually define a map $f:\mathbb{Q}[X,Y]\to\mathbb{Q}$ (...
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Maximal Ideal of Subring of K[x,y]
I came across the subring $S=K[x,xy,xy^2,\ldots]$ of $R=K[x,y]$ (where $K$ is a field) in a book about commutative algebra.
The author claims that the ideal $I=(x,xy,xy^2,\ldots)$ is maximal in $S$.
I ...
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How are the maximal ideals of a semisimple ring related to the ideals of the center of the ring?
Let $R$ be a semisimple ring, in the sense that the regular left module ${}_R R$ is semisimple. By the Wedderburn–Artin theorem I would expect there to be a bijection between the maximal (say left) ...
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Elements in a local ring written as a sum.
Set $A$ as a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ generated by $m_1,\ldots,m_n$.
I believe, every element $a \in A$ can be written as a finite sum:
\begin{align*}
a= \...
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2
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Understanding how a ring $R$ can be viewed as a ring of functions on $\operatorname{Spec} R$ using polynomial rings as an example
I've read that a ring $R$ is often viewed as a ring of functions on its spectrum $\operatorname{Spec} R$ in the following way: $f\in R$ is the function in question, $x\in \operatorname{Spec}R$ an ...
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Prime and Maximal Ideals of $\mathbb{Z}[x]$ containing $(30,x^3+1)$
What I've worked out so far is that $(30, x^3+1)$ is contained in $(2, x^3+1)$, $(3, x^3+1)$, and $(5, x^3+1)$.
For $(2, x^3+1)$, we have $\mathbb{Z}[x]/(2, x^3+1) \simeq \mathbb{F}_2[x]/(x^3+1)$. I ...
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2
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Questions of Proposition 9.2 from Atiyah's Introduction to Commutative Algebra
Proposition 9.2. Let $A$ be a Noetherian local domain of dimension one, $\mathfrak{m}$ its maximal ideal, $k = A / \mathfrak{m}$ its residue field.
Then the following are equivalent:
i) $A$ is a ...
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37
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What is the power of an ideal in the commutative unital ring R.
If I is an ideal in a commutative unital ring R, then what is the elements of $I^{n}$?
P.S. I know the definition JK, where J and K are different ideals. It's all the possible finite sum of their ...
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find a prime ideal which contains $(x^2+1,y^2+1)$
The problem is :
check $I=(x^2+1,y^2+1)$ is not a prime ideal in
$\mathbb{Q}[x,y]$ and find a proper prime ideal $I_1$ such that
$I\subset I_1$
I have checked $I$ is not a prime ideal in $\mathbb{...
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Questions of commutative algebra proposition2.8 from Atiyah
Proposition 9.2. Let A be a Noetherian local domain of dimension one, m its
maximal ideal, $ k = \frac{A}{m}$ its residue field. Then the following are equivalent:
i)A is a discrete valuation ring;
ii)...
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Conductor coprime to an ideal of a ring of integers in a number field
Let $L/K$ be an extension of number fields. Suppose that $\alpha \in \mathcal O_L$ is a primitive element, i.e., $L = K(\alpha)$. Of course, we have $\mathcal O_K[\alpha] \subset \mathcal O_L$. The ...
- 1,567
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Finding all prime ideals in $\mathbb{Z}[x,y]$ that contain $I=(275,(x^2+4)^{1729},y^{2022})$
Any prime ideal in $\mathbb{Z}[x,y]$ that contains the ideal $I=(275,(x^2+4)^{1729},y^{2022})$ must contain $(275,x^2+4,y)$, and hence must contain exactly one of $(5,x^2+4,y)$ or $(11,x^2+4,y)$, but ...
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