Articles of ideals

Show that for a field $F$, the polynomial ring $F$ is not a PID for $n>1$.

I want clarification of the following solution: Let $I=(x_1)+(x_2)$ be an ideal of $F[x_1, x_2, \ldots, x_n]$. Then if $I=(f)$ is principal then we must have $f \in F \backslash \{0\}$ since $\gcd(x_1, x_2)=1$ and $f \mid x_1,\ f \mid x_2$. Why? But $I \cap F =\{0\}$. Why? Contradiction, why?

Density of set of splitting primes

Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$. Then the claim is that the set of completely splitting primes in the extension $K(\mu_{p^r}) | K$ for every rational prime $p$ and $r […]

Ideal of $\text{End}_k V$ has certain form.

Let $V$ be a finite-dimensional vector space over field $k$ and $R = \text{End}_k V$. How do I see that any left ideal of $R$ takes on the form $Rr$ for some suitable element $r \in R$?

Class Group of Ring of Integers of $\mathbb{Q}$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need only look at the primes $2,3,5,7$. Moreover, because $x^2+57 \equiv x^2 \mod 3$, $x^2+57 \equiv x^2+2 \mod 5$, $x^2+57 \equiv x^2+1 \mod 7$, it […]

Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice. I would like to find all elements of quotient ring $\mathbb{Z}[i]/I $ , where $\mathbb{Z}[i] =\{ {a+bi|a, b∈ \mathbb{Z}}\}$ – Gaussian integers and $I$ is ideal $I = (2 + 2i) […]

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that $m\otimes n$ is a simple tensor so we have $m,n\in I$. As $I$ is principal ideal we have $m=pa$ and $n=qa$ for […]

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 […]

What are some examples of principal, proper ideals that have height at least $2$?

Krull’s principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some (preferably simple) examples where the result of the theorem doesn’t hold? A search for similar examples turned up this question, but perhaps not requiring […]

Kernel of the homomorphism $\mathbb C → \mathbb C$ defined by $x→t,y→ t^{2},z→ t^{3}$.

I think we have $z-x^3$, $y-x^2$, and $z^2-y^3$ as elements of the kernel of the homomorphism $\mathbb C[x,y,z] → \mathbb C[t]$ defined by $x→t,y→ t^{2},z→ t^{3}$. But why the kernel is not generated by all the 3 elements, and only by $z-x^3$, $y-x^2$? I think maybe it is because of $z^2-y^3$ is in $\left<z-x^3, y-x^2\right>$, […]

Extended ideals in power series ring

Let $A$ be a commutative ring with $1$ and consider the ring of formal power series $A[[X]]$. If $I \subseteq A$ is an ideal, let $I[[X]]$ denote the set of power series with coefficients in $I$. This is an ideal; it is the kernel of the reduction homomorphism $A[[X]] \to (A/I)[[X]]$. Let $IA[[X]]$ denote the […]