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

For questions about prime ideals and maximal ideals in rings.

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### Isomorphism in the quotient ring of a localization

Let $R$ be a ring with unity and suppose $I$ is a maximal ideal of $R$ such that $M = R \setminus I$ is a right denominator set for $R$. Is it true that $R_M/I_M \simeq R/I$, where $R_M$ is the right ...
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### Could someone explain the steps to finding the generator of the prime ideal

Question: Answer: So I understand part a and the fact that because it's euclidean it must be a PID but how do you get (-1,4) are the minimal co-ords is it just trail and error? and the fact that 3+4i ...
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### What is a prime ideal in $\mathbb{Z}[2x]$?

Let $R = \mathbb{Z}[2x]$, how does a nonzero prime ideal look like? An element of $R$ is of the form $$f(x) = a_0 + a_1(2x) + a_2(2x)^2 +... + a_n(2x)^n, a_i \in \mathbb{Z}.$$ Comparing with the case ...
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### A question related to the normality of a ring

Consider the ring $R=k[x^2 , xy,y^2 ]\subset k[x,y]$. My goal is to show that $R$ is normal but not a UFD. Here is what I have gathered so far: $(1)$ One can show that $R\cong k[x,y,z]/(x^2 -yz)$. ...
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### Computing the fiber over $(0,0)$ for the normalization $\operatorname{Spec} \Bbb C[t]\to\operatorname{Spec} \Bbb C[x,y]/(y^2-x^2(x+1))$

Let $R=\mathbb{C}[x,y]/(y^2 -x^2 (x+1))$. Setting $t:=y/x$, we can show that $R[t]=\mathbb{C}[t]$ and use it to conclude that $R[t]$ is the normalization of $R.$ Now consider the corresponding ...
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### A question related to contraction of prime ideals

This question is based on an example from Mel Hochster's notes on commutative algebra. Let $k$ be a field. Consider $R=k[X(1-X),Y,XY]\subset S=k[X,Y]$. It is straightforward to show that this is an ...
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### Generators of a finite intersection of maximal ideals

In the polynomial $k$-algebra $A$, in $d$ variables over a field $k=\bar k$, is it true that any finite intersection of maximal ideals is generated by $d$ elements? If $I=m_1\cap\dots\cap m_n$, with ...
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### On the preimage of maximal ideals

Let $\varphi:R\to S$ be a map of commutative rings. It is well known that if $M$ is maximal in $S$ then the preimage $\varphi^{-1}(M)$ need not be maximal in $R$ (consider $\mathbb{Z}\to\mathbb{Q}$). ...
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### Proof that an Ideal is not a prime Ideal in the commutative Ring of Complex Numbers

Let $R$ be a commutative ring and consider the ideal $I=(x^2+1)$ of the polynomial ring $R[x]$ generated by $x^2+1$. If $R=\mathbb{C}$ prove that $I$ is not a prime ideal. In order to prove that, I ...
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### Noetherian domain with unique principal prime ideal that is not a DVR

The question is whether such a thing exists. Namely, a discrete valuation ring (DVR), in whatever way you define it, is quite obviously a domain, Noetherian, and has a unique prime element up to ...
1 vote
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### Classification of associated primes of a module.

Suppose $M$ is a finitely generated $\mathbb{Z}$-module. I am interested in classifying the associated primes of $M$. I know that $M$ is an $\mathbb{Z}$-module, it is an abelian group. So I think I ...
1 vote
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### Finitely generated module over Noetherian ring, all whose localizations at associated primes of the ring is $0$, is a torsion module?

Let $M$ be a finitely generated module over a commutative Noetherian ring $R$ such that that $M_P=0$ for every associated prime $P$ of $R$. Then, is it true that for every $m\in M$, there exists a non-...
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### If $M$ is a maximal right ideal of a ring $R$ with identity element and $a \in R-M$ then $a^{-1}M$ is also a maximal right ideal.

I want to show that if $R$ is a ring with identity element and $M$ is a maximal right ideal of $R$, $a\in R-M$, then the set $a^{-1}M=\{r\in R | ar\in M\}$ is also a maximal right ideal. I already ...
1 vote
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The Chinese remainder theorem holds for Dedekind domains, and implies that if $\mathfrak{p}_1,...,\mathfrak{p}_n$ are prime ideals of a Dedekind domain $R$, and $\{a_1,...,a_n\}\subset frac(R)$, $\{... 0 votes 1 answer 80 views ### Why doesn't every ring have a maximal ideal? Hello I'm reading the proof for Theorem 7 here and I'm struggling to grasp at which point it's necessary that the ring be commutative. I tried looking elsewhere and through other posts here but am ... 3 votes 1 answer 68 views ### Ideals contained in a prime$\mathfrak p$and containing a$\mathfrak p$-primary ideal Let$R$be a Noetherian ring and$\mathfrak q$a$\mathfrak p$-primary ideal. Is it true that every ideal$\mathfrak q\subseteq \mathfrak a\subseteq \mathfrak p$is$\mathfrak p$-primary? The ... 2 votes 1 answer 40 views ### Equivalent statements of a finitely generated module being locally free Let$M$be a finitely generated$R$-module. Prove that the following conditions on$M$are equivalent: (a)$M$is locally free over$R$(i.e.$M_m$is free over$R_{m}$for all maximal ideals$m\...
I am trying to prove that the Nilradical of a ring $A$ is the intersection of the minimal prime ideals. The one inclusion is obvious but I cannot do the other. I am just taking an $x$ which belongs to ...