Articles of prime ideals

Polynomial splitting in linear factors modulo a prime ideal

Let $K$ be a number field and let $f(x) \in \mathcal{O}_K[x]$ be an nonconstant irreducible polynomial. Also let $L = K(\alpha)$ be an extension of $K$ containing a root of $f(x)$ and let $P$ be a prime of $\mathcal{O}_K$ that splits completely in $L$. My question is: Is it true that $f(x) \bmod P$ completely […]

bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$

Given a ring $R$, I want to show that the localization of $R$ at the prime ideal $P$ of $R$ (denoted as $R_P$) is isomorphic to the set of prime ideals of $R$ contained in $P$. That is: $$ \text{Spectrum}(R_P)\cong \{I\subseteq P \mid \text{$I$ is an ideal of $R$}\} $$ From the statment, I can […]

The “general algorithm” for primary decomposition

I’m trying to learn the process of primary decomposition of a monomial ideal and struggling. I can’t seem to find a resource where I’m able to follow the steps in an example and see the thinking behind them. If, say, we have the ideal $I=\langle x^2, xy, x^2z^2,yz^2\rangle$, the first step I take is to […]

Showing there is a prime in a ring extension using Nakayama's lemma

Here’s the problem that I’m working on: if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, Undergraduate Commutative Algebra, Exercise 4.12(i)) There was a hint to use Nakayama’s lemma to […]

Uniqueness of prime ideals of $\mathbb F_p/(x^2)$

What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ corresponds to a prime ideal $P$ of $\mathbb F_p[x]$ containing $(x^2)$. And then $P$ contains $(x)$ since it […]

How to prove every radical ideal is a finite intersection of prime ideals?

Let $k$ be an algebraically closed field, $\mathbb{A}^n(k)$ the affine space corresponding to $k[x_1, \dots, x_n]$, which is Noetherian by the Hilbert Basis Theorem. We know that any algebraic set $V$ can be written as the finite union of irreducible algebraic varieties, $V_1 \cup \dots \cup V_k$, no one containing the other (See Hartshorne, Algebraic […]

Spectrum of infinite product of rings

$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod \Spec(R)$. My intutition says that this does not extend to the infinite case. Maybe $\Spec(\oplus_{i \in \Z} R) = \coprod_{i \in \Z} R$ holds, but I am […]

Do there exist a non-PIR in which every countably generated prime ideal is principal?

Is there a commutative ring $R$ such that all the countably generated primes are principal, but $R$ is not a principal ideal ring? I know that if all the prime ideals are principal, then all the ideals are principal (see here ; this answer doesn’t need $R$ to ba a domain at all). On the […]

Show that $\langle 13 \rangle$ is a prime ideal in $\mathbb{Z}$

To show that $\langle 13 \rangle$ is a prime ideal in $D= \mathbb{Z[\sqrt{-5}]}$, I could show that $13$ is an irreducible element of $D$ but as $D$ is not a U.F.D, it is not of much use I guess. How can I prove this.

An example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

I am looking for an example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime. In a Prüfer domain such an intersection is always a prime ideal.