Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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Proving if the ideal $\mathfrak{a}=(Y+2X^2,Z+3X^3,T^5-X^4-Y-Z)$ is prime or not in $K[X,Y,Z,T]$.

Note: I will abuse of notation and write $n$ instead of $n\cdot1$, for any $n \in \mathbf{Z}$. Obviously, $1$ being the identity of $K$. Applying the first isomorphism theorem and the correspondence ...
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Let $M$ be an $R$-module. If $I \subseteq \mathfrak{m}$ implies $M_\mathfrak{m} = 0$ for all max ideals $\mathfrak{m} \subseteq R$, show $IM = M$

Suppose we have a ring $R$ and an $R$-module $M$. Suppose we have an ideal $I\subseteq R$ such that for all maximal ideals $\mathfrak{m} \supseteq I$ we have $M_\mathfrak{m} = 0$. I need to show that ...
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$(x^2-5)$ is not a prime ideal of $\mathbb{R}[x]$

This question might be very naive but I still want to varify if my idea is right, since I am only beginning ring theory. I have to show $(x^2-5)$ is not a prime ideal of $\mathbb{R}[x]$. My thought is ...
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Show that $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$ [duplicate]

I want to show that the ideal $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$. I know that $\mathbb{Z}/(2)[x]$ is isomorphic with $\mathbb{Z}[x]/(2)$ and also know that since $\mathbb{Z}/(2)[x]$ is ...
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Non-nilpotent Elements Not Contained in Every Prime Ideal, But What About All Maximal Ideals [duplicate]

I am currently studying for an algebra qualifying exam and am stuck on the second part of a problem. Assume R is a commutative ring (it doesn't specify if the ring has a multiplicative identity). The ...
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Determining if an ideal is prime / determining if the quotient ring is an integral domain. [duplicate]

Consider $p(x)=3x^2+x+2$. In a prior exam assignment, one of the tasks was to determine if $\mathbb{Z}_2[x]/(p(x))$ is an integral domain, with $(p(x))$ being the ideal generated by $p(x)$. A theorem ...
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Show that $R\cong R_P$, the ring of quotients of $R$ with respect to the multiplicative set $R-P$ if $R$ has exactly one prime ideal $P$.

Question: Let $R$ be a commutative ring with identity that has exact one prime ideal $P$. Show that $R\cong R_P$, the ring of quotients of $R$ with respect to the multiplicative set $R-P$ This is an ...
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Is the ideal, prime? maximal? Is it principal?

im sitting with this exercise: Let $J$ be the ideal of $\mathbb Q[x]$ generated by two polynomials: $f(x)=(x+1)(x+2)(x^2+1)$ and $g(x)=(x+1)^2(x+2)^2$. Is the ideal $J$ prime? Is it maximal? Is it ...
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Characterization of maximal multiplicatively closed subsets

Let $A$ be a non-zero ring, and let $\Sigma$ be the set of the multiplicatively closed subsets properly contained in $A$. Show that $\Sigma$ admits a maximal element respect to inclusion. Then prove ...
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Question about a proof involving local rings: $R$ has exactly $3$ ideals, show that if $a,b\in I$ then $ab=0$

I have a question regarding the following thread: Commutative unitary ring with exactly three ideals. I believe I've put the pieces together, but I am, for whatever reason, feeling uncomfortable. So, ...
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Exercise with localization and minimal primes

Let $A$ be a ring and $p\subset A$ be a prime ideal. Call $f$ the canonical map $A\to A_p$, and set $I:=\operatorname {ker }f$. Show that $I\subseteq p$ and that $\sqrt I=p$ if and only if $p$ is ...
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A prime ideal of a polynomial ring over a PID can be generated by two elements. [duplicate]

I am studying for a qualifying exam, and have been working on this problem: Let $D$ be a PID and let $P$ be a prime ideal of the polynomial ring $D[x]$. Suppose that $P$ contains a non-zero constant ...
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If $p$ is a prime in $\mathbb{Z}$ and $(p)$ is a prime ideal in $\mathbb{Z}[i]$, then $x^2 \equiv -1\pmod{p}$ has no solution in $\mathbb{Z}[i]$

I can't find the proof for this. I know that $\mathbb{Z}[i]/(p)$ is an integral domain, so I tried to assume that there is a solution for $x \in \mathbb{Z}[i]$ and reach a contradiction of that fact, ...
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Let $R$ be a commutative ring with $1$ that is not a field. Prove that $R[x]$ is not a PID. [duplicate]

Question: Let $R$ be a commutative ring with $1$ that is not a field. Prove that $R[x]$ is not a PID. Thoughts: I am seeing a lot of questions around SE about this, but most are showing that if $R[x]$...
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Proving $\pi$ is irreducible/prime in $R.$

Can I please get feedback on my proof or help proving the following problem I am working on? Thank you for your time and help. Denote $R$ as the ring of algebraic integers in an imaginary quadratic ...
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If $\mathfrak p$ is a prime ideal of an integral domain, is $\mathfrak pM$ a prime submodule of a torsion-free module $M$?

Let $M$ be a torsion-free module of an integral domain $R$. A submodule $N$ of a module $M$ is said to be prime if for all $r\in R$ and $m\in M$, $rm\in N$ implies either $rM\subseteq N$ or $m\in M$. ...
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Intersection of associated prime ideals is radical of annulator

As an exercise I have to prove: Let $M$ be a finitely generated $R$-module, where $R$ is a noetherian ring (commutative and with $1$). Then: \cap_{\mathfrak{p} \in \text{Ass}(M)} \mathfrak{p} = \...
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Showing that $A/J \cong \mathbb C.$

Consider the $C^{\ast}$-algebra $A = C_0(\mathbb R)$ and the ideal $J = \{f \in A\ |\ f(0) = 0 \}$ of $A.$ Show that $A/J \cong \mathbb C$ as $C^{\ast}$-algebras. Here $J$ is clearly a closed maximal ...
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Suppose that the ring $A$ has only a maximal ideal $\mathfrak m$ and that $\mathfrak m$ is principal (denote its generator with $t$). Assume also that $\bigcap_n \mathfrak m^n=0$. I must show that ...
When and why does $r|m \Rightarrow (p^r - 1)|(p^m-1) \Rightarrow (T^{p^r}-T)|(T^{p^m}- T)$ hold? [duplicate]
I've come across a proof where they use the fact that for $p$ prime, $r, m \in \mathbb{N}$ such that $r|m$ it holds that $(p^r - 1)|(p^m-1)$. I've tried to understand why this must be true, but haven'...
Prime ideal in $\mathbb{Z}[x,y]$ containing another ideal
I am preparing for my algebra exam, and I am stuck with this problem: Find all maximal and prime ideals in $\mathbb{Z}[x,y]$ containing the ideal $\mathbb{I}=(55,x^2+4,y).$ What is the general ...