# Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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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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### 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 ...
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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 ...
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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 $A$. ...
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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 ...
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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 ...
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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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1 vote
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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 ...
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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 ...
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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 $M$ ...
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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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### 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 ...
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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 ...