# 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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### Show that $R$ is a field

Let $R$ be a commutative ring with unit. If $R\neq 0$ such that each finitely generated $R$-module is free then $R$ is a field. In my notes there is the following proof: We need to show that ...
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### In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
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### Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$ [duplicate]

I have a ring $\mathbb Z[\sqrt{-2}]$ and I need to describe all the prime numbers of that ring. How I can do that? Thank you
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### Polynomial with infinite roots

I started Ring theory recently and I came across this statement while reading polynomial rings.. If $F$ is an infinite field and let $f(x)\in F[x]$ . If $f(a)=0$ for infinitely many elements $a$ ...
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### Each automorphism is of that form

Let $R$ be a commutative ring and $c,b\in R$ with $c$ invertible. The correspondence $x\rightarrow cx+b$ defines an unique automophism of $R[x]$ that is the identity in $R$. If $D$ is an integral ...
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### If $a$ and $b$ are elements in a ring with $a^n=b^n$ and $a^m=b^m$ then $a=b$

I was doing the first exercises from the book Exercises in Basic Ring Theory by G. Călugărescu and P. Hamburg and I found one whose solution isn't quite clear to me. Ex. 1.4 If $a$, $b$ are ...
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### Exercise on the ring $\mathbb Z \times \mathbb Z$ and its quotient with an ideal

Let $A = \mathbb Z \times \mathbb Z$ a ring, where operations are defined elementwise. a) Prove that the ideal $I$ generated by $x = (4,6)$ is not maximal. b) Find in $A$ (if it exists) an ...
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### Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
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### Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $\pi\colon S\to \mathbb{C}$ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
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### Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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### Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
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### $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors

I want to show that $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors: given a chain $a_1,a_2,\dots,a_n,\dots$ and $a_{n+1}\mid a_n$ for any $n\in \mathbb{N}$, then there is ...
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### $I$ is the maximal left ideal

Let $R$ be a ring and $I\subseteq R$ the unique maximal right ideal of $R$. I have shown that $I$ is an ideal and that each element $a\in R-I$ is invertible. I want to show that $I$ is the unique ...
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### Is $a$ invertible? [closed]

We have that $R$ is a ring. Suppose that $Ra=R$ and $bR=R$, for $a,b\in R$. Then we have that there is $x\in R$ such that $ab=1$ and $bx=1$. Does it follow that $a$ is invertible?
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### Simple examples of rings from topology

The ring $C([0,1],\mathbb{R})$ of continuous functions from $[0,1]$ to $\mathbb{R}$ is an interesting example of ring due to its some interesting property (namely, structure of maximal ideals). The ...
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### Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: \sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
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### Showing an isomorphism of rings

Consider the ideal $I=(1+2x)\cdot \Bbb Z[x]$ in the polynomial ring $\Bbb Z[x]$. I am trying to show that $\Bbb Z[x]/I$ is isomorphic to $R=\{\frac{a}{2^r}:a\in \Bbb Z, r\in \Bbb N_0\}$. My approach:...
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### $R/\langle p^k\rangle$ is an associator (i.e. if $\langle a\rangle = \langle b\rangle,$ then $a$ and $b$ are associates) when $R$ is a PID.

As the title says, I want to show that when two principal ideals are equal in $R/\langle p^k\rangle,$ where $R$ is a principal ideal domain and $p\in R$ is a prime element, then their generators are ...
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### Meaning of $Z\oplus Z$

I am a beginner in Ring Theory and just started Integral Domains. In my textbook, the following was stated : $Z\oplus Z$ is not an integral domain. I can't understand this. I know $\oplus$ ...
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### Can I deduce $f(\textrm{Ker}(g))=g(\textrm{Ker}(f))=0$ from this data?

Let $R$ be a commutative ring with identity and $A, B$ two $R$-algebras. Consider $f, g: A\longrightarrow B$, $h:B\longrightarrow A$ and $\imath:A\longrightarrow A$ morphisms of $R$-algebras ...
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### Proving that there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$ [duplicate]

Let $I_1,...,I_m$ be ideals of a ring $R$ such that $I_j+\cap_{k\neq j}I_k=R$ for every $j\in\{1,...,m\}$. Then if $a_1,...,a_m\in R$ there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$ for ...
I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate ...
### Proving/Disproving $M$ has the structure of an $R$-module
Given an abelian group $M$ and a ring $R$, how can one prove or disprove that $M$ has the structure of an $R$-module? When proving $M$ is an $R$-module, if it is not obvious how to define an action \$R\...