Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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$ \mathfrak mR_{\mathfrak m} $-primary ideal is the localization of some $\mathfrak m$-primary ideal?

Let $\mathfrak m$ be a maximal ideal of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R_{\mathfrak m}$ such that $\mathfrak m^n R_{\mathfrak m}\subseteq J \subseteq \mathfrak mR_{\...
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Prime ideal of $\mathbb{C}[x,y]$.

I want to show that $J:= \langle x-y^2+1 \rangle$ is a prime but not a maximal ideal of $\mathbb{C}[x,y]$. My idea is that $x-y^2+1=x-(y-1)(y+1)$ and so $\langle x-y^2+1 \rangle \subsetneq \langle x, ...
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3 answers
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How to determine if the ideal $I = \langle x-1, y \rangle$ is a maximal ideal of $\mathbb{Q}[x, y]$

How do I determine whether $\mathbb{Q}[x, y] / I$ is a field? Where I is generated by the Gröbner basis $$ I = \langle x-1, y\rangle = \bigl\{ a(x,y)\,(x-1) + b(x,y)\,y \mid a,b \in \mathbb{Q}[x,y] \...
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Is $\bigcap_{n=1}^\infty I^n$ contained in a minimal prime ideal?

Let $I$ be a proper ideal of a commutative Noetherian ring $R$. Let $J:=\bigcap_{n=1}^\infty I^n$. Then, is it true that $J \subseteq P$ for some minimal prime $P$ of $R$? By prime avoidance, I am ...
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Describe $\operatorname{Spec}(\mathbb{R}[x]/(x^n))$

I want to describe $\operatorname{Spec}(\mathbb{R}[x]/(x^n))$. For starters, I know that since $\mathbb{R}$ is a field, then $\mathbb{R}[x]$ is a PID, and therefore the only prime ideal in $\mathbb{R}[...
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Does $\{(f):ℙ↣ℙ\}$ contain any analytic members $f$?, and if so What is the simplest such injective $f$? Are all $f$ necessarily monotonic? [duplicate]

(Above, $ℙ≔\{\text{all primes}_ℤ\}$.) Are there any analytic functions that will give a unique prime output for every distinct prime input? Analyticity should preclude cheap reiterating upon a prime-...
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Given the ring $R:=(\mathbb{Z}/n\mathbb{Z})$ find its maximal ideals $M$ and the number of elements in $R/M$ [duplicate]

I recently stumbled across this question for the rings $R:=(\mathbb{Z}/8\mathbb{Z})$ and $R:=(\mathbb{Z}/30\mathbb{Z})$ and I was wondering wether this could be generalized for any given ring $R:=(\...
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Giving the size of the quotient subring $(\mathbb{Z}/30\mathbb{Z})/I$ where I is a maximal ideal [duplicate]

I need help with the proof of a specific step in the following problem: Given the ring $\mathbb{Z}/30\mathbb{Z}$ give the size of the quotient subring $(\mathbb{Z}/30\mathbb{Z})/I$ where I is a ...
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Prime ideals of subrings of the ring of Gaussian integers

Can anyone give me a hint with proving the following, Let $\alpha\in\mathbb Z[i],$ and let $P$ be a non-zero prime ideal of $\mathbb Z[\alpha].$ Show that the quotient $\mathbb Z[\alpha]/P$ is a ...
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1 answer
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Sets without Dirichlet/polar density

Let $K$ be a number field with norm $N=N_{K/\mathbb{Q}}$, $\mathcal{P}$ its set of prime ideals, and $A\subset\mathcal{P}$. We say that the Dirichlet density is $$\delta^D(A):=\lim\limits_{s\to1^+}\...
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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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1 vote
1 answer
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Find a sequence in a non-principal ultrafilter that decreases to a point

Let $\mathcal F$ be a non-principal ultrafilter of subsets of the set $X$. Let $x \in X$. Does there exist a decreasing sequence $F_1 \supset F_2 \supset ...$ in $\mathcal F$ such that $\bigcap_n F_n ...
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The only maximal ideal of the set of all function germs around $p$

Here is the definition we are using for the set of all function germs around p: Now, I want to show that $m(p) := \{\bar{\phi} \in \mathcal{\varepsilon}(p)| \bar{\phi}(p) = 0\}$ is the only maximal ...
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How to approach proof of finding number of prime ideals lying over another prime ideal

Consider the following problem: Problem: Let $f= f(X) \in \mathbb{Z}[X]$ be a monic polynomial of positive degree and let $B=\mathbb{Z} [X] /\left<f\right>$. For a prime number $p$, show that ...
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1 answer
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Understanding Prime Ideal [duplicate]

Let $I$ be a prime ideal of ring $R$. I know $ab\in I$ implies $a\in I$ or $b\in I$. Does $0\in I$? If not, when is this assertion valid?
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Why is this residue field equal to the field of fractions in this case? [duplicate]

I have the following question. If $p$ is a prime ideal then by localization with the multiplicative set $S:=R\setminus p$ we get the local ring $R_p=:S^{-1}R$ and a maximal ideal $pR_p$. By ...
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If factor ring is domain ( non commutative ), then quotient must be prime

Let $\frac{R}{P}$ be a domain ( not necessarily commutative ). Show that $P$ is prime. In previous questions asked on this site, integral domain ( commutative ) is assumed, but I don't think this is ...
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1 vote
1 answer
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Prime ideal $\implies$ maximal in a Boolean ring

I want to show that a prime ideal in a non-unital Boolean ring $B$ is maximal ideal. If the ring contains unity then it is easy. As Boolean rings are commutative, for a prime ideal $P$ the ring $B/P$ ...
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1 answer
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A question related to the codimension of ideals

I am currently working on a proof of the following statement Let $R$ be a ring and suppose $U \subset R$ is multiplicatively closed. Let $S:=U^{-1}R$. If $P\subset S$ is a prime ideal, then the ...
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If $A$ is Noetherian, and $x$ is a non-unit and non zero-divisor, then can $(xr)=(r)$ for any non-unit $r$?

Problem I am attempting to prove that if $A$ is a Noetherian commutative ring with unit, and if $x$ is a non-unit and not a zero divisor, then any minimal ideal of $(x)$ has height $1$. Attempt I have ...
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3 answers
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Is $(2,x,y)$ maximal ideal in $\mathbb Z[x,y]$?

I’m new to ring theory and have difficulty in showing whether an ideal $I$ is a maximal ideal of commutative ring $R$ with unity mostly when $I$ is generated by more than one element. For example: The ...
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4 votes
1 answer
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Codimension inequality of prime ideal in a regular local ring

The following is an exercise (#6) of Eisenbud's commutative algebra, chapter 10: Exercise. We mentioned that if $P$ is a prime ideal in a regular local ring $R$ and if $R\to S$ is a map of local rings,...
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-1 votes
1 answer
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Primes in the support appear in prime chains?

Let $M$ be a finitely generated module over a Noetherian ring $A$. Suppose that $p\subseteq A$ is a prime ideal which is contained in the support of $M$, i.e., $M_p\neq 0$. Then does there always ...
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Factor ring of Gaussian integers is a field

I want to show that $E=\mathbb{Z}[i]/\langle2-i\rangle$ is a field. To do this, I note that $R/I$ is a field iff $I$ is a maximal ideal. Moreover, a maximal ideal is a prime ideal in a commutative ...
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1 answer
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Existence of a prime ideal containing all but one minimal prime ideals

Does there exist a prime ideal containing all but one minimal prime ideal in an unital commutative Noetherian ring $A$, for any given minimal prime ideal $\mathfrak{p}$? I encountered this statement ...
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Show that $\sigma$ induces an injective ring homomorphism $R/\sigma^{-1}(\mathfrak{m}) \to S/\mathfrak{m}$.

Let $\sigma: R \to S$ be a ring homomorphism and $\mathfrak{m}$ a maximal ideal of $S$. Show that $\sigma$ induces an injective ring homomorphism $R/\sigma^{-1}(\mathfrak{m}) \to S/\mathfrak{m}$. Is ...
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1 vote
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Is every parameter outside of the union of minimal primes?

Proposition 1: Let $R$ be a commutative, Noetherian ring and $\ \mathfrak{p} \in \operatorname{Spec}(R)$. If $\operatorname{ht}(\mathfrak{p})=h$, then there exist $y_1, \ldots, y_h \in R$ such that $\ ...
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1 answer
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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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0 answers
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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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2 votes
1 answer
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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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3 votes
2 answers
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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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2 votes
1 answer
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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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0 answers
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A question about the proof $\dim(k[x_1,\dots,x_n]) = n$

I am working on a proof of the Krull dimension of $K[x_1,\dots, x_n]$ for $K$ a field. I am stuck on the following step. Suppose that $\dim(K[x_1,\dots, x_l]) = l$ and that the maximal chains of $k[...
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1 vote
0 answers
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Number of roots in cyclotomic polynomial $\Phi_{15}[x]$ in $\mathbb F_p$

I'm trying to understand why if the $gcd(p-1, 15) = d \neq 15$, then there are zero roots (since if it's $=15$, there are exactly 8). I was thinking that since a solution to $x^d - 1$ is relevant if $...
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1 vote
1 answer
69 views

If $I$ is an invertible ideal, then there is $\alpha$ with $N(\alpha I^{-1})$ coprime with $N(I)$

Let $\mathcal{O}$ be an order inside a quadratic number field (not necessarily maximal). I want to show that if $I$ is an invertible $\mathcal{O}$-ideal, then there is $\alpha \in I$ such that $N(\...
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0 answers
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Ideals containing $\langle h,y \rangle \subset \mathbb{C}[x,y]$, with $h \in \mathbb{C}[x]$ separable

Let $h \in \mathbb{C}[x]$ be separable of degree $d \geq 2$, in other words, there exist distinct $c_1,\ldots,c_d \in \mathbb{C}$ such that $h=(x-c_1)\cdots(x-c_d)$. Let $I:=\langle h,y \rangle \...
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3 votes
1 answer
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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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2 votes
0 answers
21 views

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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0 votes
1 answer
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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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5 votes
1 answer
165 views

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 ...
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1 vote
0 answers
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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 ...
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  • 284
1 vote
1 answer
50 views

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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  • 129
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0 answers
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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 ...
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1 vote
0 answers
28 views

Strong approximation for maximal orders in central division algebras

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 $\{a_1,...,a_n\}\subset frac(R)$, $\{...
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  • 251
0 votes
1 answer
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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 ...
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  • 349
3 votes
1 answer
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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 ...
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2 votes
1 answer
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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\...
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0 votes
0 answers
32 views

Nilradical is the intersection of the minimal prime ideals [duplicate]

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 ...
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1 vote
0 answers
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$M_p \cong N_p$ for all prime ideals $p$, but $M \not \cong N$

I am asked to find an example of a ring $R$ and $R$-module $M$ and $N$ such that $M_p \cong N_p$ for all prime ideal $P$ in $R$ but $M$ is not isomorphic to $N$. My idea is as follows: Recall that if $...
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2 votes
1 answer
30 views

Elements of a ring determined by values mod primes

In $\mathbb{Z}$, or more generally a Dedekind domain $D$, an element $x$ is uniquely determined by its image in the associated prime fields $D/\mathfrak{p}D$ as $\mathfrak{p}$ varies over all prime ...
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