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

For questions about prime ideals and maximal ideals in rings.

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### A variety $X$ is irreducible $\iff I(X)$ is prime

EDIT: after I finished writing all this I came up with a possible solution, I decided to post it anyways (hope it's not against the rules) for two reason: This may be useful to other people with the ...
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### $0 \subset (x_1 - a_1) \subset (x_1 - a_1, x_2-a_2) \subset … \subset (x_1-a_1,…, x_n - a_n)$ is a (strictly) ascending chain of prime ideals

I saw the following exercise in Sharp's text book: Let $K$ be a field, and let $R = K[x_1,...,x_n]$ be the ring of polynomials over $K$ in indeterminates $x_1,...,x_n$; let $a_1,...,a_n \in K$. ...
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### Prove that $\sqrt{I} = \sqrt{J} \Leftrightarrow V(I) = V(J)$

Question: Let $I$ and $J$ are ideals of a commutative ring $R$. Prove that $\sqrt{I} = \sqrt{J} \Leftrightarrow V(I) = V(J)$, in which $V(I) = \{P \in \text{Spec}(R)\ |\ I \subseteq P\}$. We know ...
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### Which are the possible maximal ideals in $C[0,1]$ containing a prime which is not maximal?

Let $A=C[0,1]$. We know that the maximal ideals of $A$ are of the form $$M_α=\{f∈A \mid f(α)=0\}, \ α∈[0,1].$$ Now we show that there is a prime ideal $P$ which is not maximal in $A$. Consider $S$ ...
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### Prime ideals after extension of scalars

Let $k$ be a field, and let $R$ be a commutative unital $k$-algebra. Let $K/k$ be a field extension. I know that if $I \subset R$ is an ideal, then $I \otimes 1 \subset R \otimes_k K$ is an ideal. ...
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### Commutative unitary ring without maximal ideal without axiom of choice

I want to find a semi-constructive example of a unitary commutative ring without any maximal ideals assuming that axiom of choice is incorrect and/or a model of $\sf ZF$ where we have such a concrete ...
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### Prove $2$ is not a prime in the ring $\mathbb{Z}[\sqrt{-3}]$

I initially looked at this and believed it could be a fairly simple proof. I started by stating that if $2$ is not a prime, then it can divide the product of $2$ elements of this ring, but cannot ...
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### Maximal ideals in $\operatorname{End}_A(M)$

$\newcommand\End{\operatorname{End}}$Let $A=k[x_1, \dotsc, x_n]$ and $M$ be a graded $A$-module. Let $E=\End_*(M)$ be the graded endomorphism ring of $M$ (i.e., $E_i$ consists of all degree-$i$-...
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### Prove that a maximal ideal is radical.

Let $m$ be a maximal ideal of commutative ring $R$. Prove that $m$ is radical. I understand that $m$ is maximal if it is proper and there are no other ideals (except $R$) that properly contain it. ...
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### Locally Noetherian Domain With Finitely Many Prime Ideals

Let R be a domain with finitely many prime ideals such that the localization at each prime, $R_{\mathfrak p}$, is Noetherian. Then is $R$ necessarily Noetherian?
I don't know how to find all zero divisors for polynomials in several variables. For example: $\mathbb{Z}_2[X,Y]/(X^2,XY,Y^2)\quad$ or $\quad \mathbb{Z}_4[X,Y]/(X^2,Y^2-XY)$ Can we to proceed ...