# Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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### Integral closure of $R[x]$ in its field of fractions over R

I feel like this might have been discussed before but I couldn't find it so I apologise if this is a very common question. If $S$ is a ring and we have a subring $R$ and an element $x\in S$ ...
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### Length of the primary component of $(xy, y^2)$ at the origin is $1$. [on hold]

As the question title suggests, how do I see that the length of the primary component of $(xy, y^2)$ at the origin is $1$?
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### How to show that the field of fractions of a domain $A$ is $\{ab^{-1}: a \in A, b \in A-\{0\}\}$?

Let $A$ be a domain. Suppose that for all $a,b\in A-\{0\}$, $Aa \cap Ab \neq \{0\}$ and $aA \cap bA \neq \{0\}$. How to show that the field of fractions of $A$ is $\{ab^{-1}: a \in A, b \in A-\{0\}\}$...
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Let $A=\mathbb{Z}[X,Y]/(Y^2-6X^2), B=\mathbb{Z}[X,T]/(T^2-6)$ where $X,Y,T$ are variables and let $x,y$ be the cosets of $X,Y$ in $A$ whilst $x',t$ be the cosets of $X,T$ in $B$. Consider the ideals ...
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### Is the group of units in $\mathcal{O}_{\mathbb{Q}(\sqrt{2})}$ finite and cyclic?

Let $K=\mathbb{Q}(\sqrt{2})$ and let $\mathcal{O}_K$ be its ring of integers. Consider the group of units in $\mathcal{O}_K$. Is it finite? Is it cyclic? My thought: For any $\alpha = a+b\sqrt{2}$, ...
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### Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...
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### What does the $(m,n)$ mean in context of $\mathbb{Z}_{(m,n)}$

On the section about the Hom sets of modules, Hungerford has an exercise that asks to show that $$\operatorname{Hom}(\mathbb{Z}_m, \mathbb{Z}_n) \cong \mathbb{Z}_{(m,n)}$$ and then in the next ...
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### Unclear proof of a proposition on semisimple rings in Lang

I've been reading through Lang's Algebra chapter about semisimple modules and semisimple rings, and after Lang proves a main structure theorem about semisimple rings he proves the following ...
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### Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
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### What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
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### $\Bbb Z[\sqrt{-5}]$ is not a PID [duplicate]

I want to show: In a PID $R$ two elements $a,b\in R$ always have a greatest common divisor. Therefore $\Bbb Z[\sqrt{-5}]$ is not a PID. For the first part: $I=\{ax+by:x,y\in R\}$ is an ideal, ...
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### Is $\Bbb Z[i]$ a Euclidean ring? [duplicate]

Is $\Bbb Z[i]$ a Euclidean ring? If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?
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### Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
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### Left- and right-sided principal ideals have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
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