Questions tagged [maximal-and-prime-ideals]
For questions about prime ideals and maximal ideals in rings.
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Every Prime Ideal in Z[x] which is not principal contains a prime?
I am working on an old qualifying exam problem, and I'm stuck. Here's the question:
Show every prime ideal $P$ in $\mathbb{Z}[x]$ which is not principal contains a prime number.
I am aware that every ...
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Why are prime ideals proper?
As children we all learn this erroneous definition of a prime number: “a number $n\in \Bbb N$ is prime iff it’s only divided by one and itself”. Well that’s fine until the teacher asked us for ...
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Completion of primary ideal is primary
I have the following question.
Suppose $(R, \mathfrak{m})$ is a local Noetherian ring, $\mathfrak{a}$ is $\mathfrak{p}$-primary ideal. Is it true that $\hat{\mathfrak{a}}$ is a primary ideal in $\hat{...
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Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal?
Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal? I read that if a polynomial is primitive then the generated ideal is prime (Why?). So $(f(x))...
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How to break symmetry of a polynomial ideal to simplify Groebner basis?
I have an ideal $I$ generated by a set of polynomials $\{ p_i \}$. There are some variable permutations to which the ideal is symmetric. By this I mean (apologies if there is a standard term for this) ...
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Is $I_p=(p,x^2+1)$ a prime ideal of $\mathbb Z[x]$? What is the maximal ideals of $\mathbb Z[x]$ containing $I_p$ where $p=2,3,5$?
Let $I_p$ be the ideal of $\mathbb Z[x]$ generated by $p$ and $x^2+1$.
Problem: Is $I_p=(p,x^2+1)$ a prime ideal of $\mathbb Z[x]$? What is the maximal ideals of $\mathbb Z[x]$ containing $I_p$ where $...
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not irreducible but prime in a non-domain [duplicate]
Consider $\mathbf Z/6\mathbf Z$ as a ring.
It is not an integral domain since it contains zero-divisors, such as the element $[3]$ for example. Note that $[3]$ is not irreducible ($[3]^2 = [3]$), yet $...
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Equivalent definition for minimal ideals for commutative rings
Background
The following post on minimal ideals is a continuation and a counterpart to the following post on maxmial ideals. The quoted materials are taken from the following sources;
Fundamentals of ...
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Question about showing $(x,y)$ is a maximal ideal of $\Bbb{Q}[x,y]/F[x,y]$ [duplicate]
Background
Theorem 1: Let $M$ be an ideal in a commutative ring $R$ with identity $1_R$. Then $M$ is a maximal ideal if and only if the quotient ring $R/M$ is a field.
Exercsie 1: Prove that $(x)$ ...
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Example of a PID with $n$ Maximal ideals
For every $n\in \Bbb{N}$, can we find a PID with exactly $n$ Maximal ideals?
This question was asked to me in an interview. They told me to use localisation but I still don't have any idea about it. I ...
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Prove that a maximal ideal is also a prime ideal [duplicate]
If $S$ is a multiplication closed subset of a ring $R$, suppose $P$ is the maximal element of the set of ideals and is disjoint from S, then $P$ is the prime ideal. I'm a little bit stuck in step 2:
(...
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...if $\mathfrak{a}\subset\cup_{i=1}^{s}\mathfrak{p_i}$, then $\mathfrak{a_1}\subset \mathfrak{p_i}$ for some $i$
The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg
Background
Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
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Understanding Radicals as Intersections of Prime Ideals and as the Preimage of a Nilradical
I ran into a proposition that the radical $r(a)$ equals the intersection of the prime ideals containing $a$. Then it is said that the latter can be understood through the following result: Let $R$ be ...
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There can't only be non-homogenous prime ideals between two homogenous primes. (Vakil 12.2.G, part c)
I'm stuck and looking for advice on part c of Question 12.2.G of Vakil's FOAG. The question states:
(a) Suppose $X \subset \mathbb{P}^n$ is an irreducible projective $k$-variety. Show that the affine ...
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Associated Prime containing non-regular element
I'm struggling with an exercise about associated primes.
Let $M$ be an $R$-module, and $a\in R$ be a non-regular element of $M$ (that is, $a$ is such that $m\mapsto am$ is not injective). Show that ...
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