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

For questions about prime ideals and maximal ideals in rings.

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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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### Prime Ideals and Integral Domains - Intentionally Constructed or Purely Coincidental?

This question is made purely out of personal curiosity. The motivation behind asking this question: Normal subgroups were first truly utilized by Évariste Galois with one of the intents being to ...
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### Is this true? $\alpha\in$ Prime Ideal $P$ if and only if $\alpha r\in P$ for all $r\in R$.

Claim. Let $P\subset \text{commutative rng}\ R$ denote some arbitrary prime ideal. Then $\alpha\in P$ if and only if $\alpha r\in P$ for all $r\in R$. Proof. If $\alpha\in P$, then by definition of ...
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### A commutative ring with unity which is also reduced having exactly two minimal prime ideals.

Let $R$ be a commutative reduced ring with unity. I want to find an example of such a ring which contains exactly two minimal prime ideals. For example if we take $\mathbb{Z_6}$ then $\langle2\rangle$ ...
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### Prove there exists a prime ideal whose elements are all zero divisors [duplicate]

Let $R$ be a commutative ring with identity and let $S$ be the set of all elements of $R$ that are not zero divisors. Show that there is a prime ideal $P$ such that $P \cap S$ is empty. The hint that ...
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### Prime Ideals of $\Bbb Z[x]$ containing $\mathbb(3, x^2+3x+5)$ [duplicate]

I need to find the prime ideals of $\mathbb{Z}[x]/(3, x^2+3x+5) \cong \mathbb{Z}_3[x]/(x^2+3x+5) \cong \mathbb{Z}_3[x]/(x^2+2)$. Therefore, we have that $(x+2)$ and $(x+1)$ are the two maximal ideals ...
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### Prove that an ideal has height 2

I know this question has been asked before, and I throughly looked at all the answers, but can't find one that matches my math knowledge, or the contents of the course the exercise is derived from. In ...
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### In $\mathbb{C}[x,y]$: When $\langle u,v \rangle$ is maximal and when $\langle u,v,w \rangle$ is maximal?

First step: Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$. Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$. $I$ is not a ...
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### $I=r(I)$ iff $I$ is an intersection of prime ideals

I am trying to solve this problem. Let $I$ be a non trivial ideal of a ring $A$, and let $r(I)$ be its radical. Prove that $I=r(I)$ iff $I$ is an intersection of prime ideals. I got the right to ...
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### Every subring of a field is an integral domain

When I was trying to prove the following theorem: Theorem. Let $E$ be an extension field of the field $F$ and let $a \in E$ .If $a$ is algebraic over $F$, then $F(a) \cong F[x]/ \langle p(x)\rangle$ ,...
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### Examples of PM rings

Def. A ring R is called a pm ring if each prime ideal is contained in exactly one maximal ideal. I asked AI to give some examples of pm rings. The answer is: Examples of PM rings in algebra include: ...
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### Is $X^3+2Y^2+3\in \mathbb{Q}[X,Y]$ prime?

I want to show that the polynomial $f=X^3+2Y^2+3\in \mathbb{Q}[X,Y]$ is prime. Since, however, $\mathbb{Q}[X,Y]$ is not a principal integral domain (as $\mathbb{Q}$ is not a field) it does not suffice ...
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### Fault in the proof that radical prime implies ideal is primary.

I have found examples regarding how the radical of an ideal being prime might not imply that the ideal itself is primary. However I am having trouble finding the error in the following proof- let $I$ ...
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### Let $R$ be a Noetherian commutative ring with zero nilradical and with any localization at a maximal ideal as a finite ring. Prove that $R$ is finite

Suppose that $A$ is a Noetherian commutative ring such that: (1) the nilradical (intersection of all prime ideals) of $A$ vanishes, and (2) localization at every maximal ideal is a finite ring. Prove ...
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Serge Lang, Algebra: Chapter 2, Exercise 1. Let $S$ be a multiplicative subset of a commutative ring $A$, containing 1 but not 0. Let $\mathfrak{p}$ be a maximal element in the set of ideals of $A$, ...