# 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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### Definition of Irreducible polynomial in terms of the unit of Integral domain.

Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product ...
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### How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD?

How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD? Moreover, how do I prove that $(7,3+\sqrt{19})$ is not a principal ideal? This is the first time I'm dealing with a quadratic integer ring ...
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### Prime elements with the same norm in a Euclidean domain [closed]

Does anybody know whether two prime elements with the same norm in a Euclidean domain are necessarily associated? Any help will be very welcome. UPD 1: It was shown that $2\pm i$ are both primes ...
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### what kind of integral domain do the non-infinite surreals form?

https://en.wikipedia.org/wiki/Integral_domain mentions the following chain of inclusions: Principal Ideal domains $\subset$ Unique Factorization domains $\subset$ GCD domains $\subset$ Integrally ...
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### quick question on ascending chain condition for rings

I know that if $R$ is a commutative ring with an identity in which every ideal if finitely generated then it satisfies the ascending chain condition. Just wondering if the converse is also true?
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### Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
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### If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is $R$ finite?

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is it true that $R$ is finite ? (I know that there are infinite domains with unity, ...
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### $D$ be a UFD, if an element of $D$ is not a square in $D$ then is it true that, that element is not a square in the fraction field of $D$?

Let $D$ be a UFD, let $F$ be the field of fractions of $D$, let $a \in D$ be such that $x^2 \ne a, \forall x \in D$. Then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is motivated ...
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### Polynomial ring, ideals and Spec

Morning everyone, I want some hint about this. i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain iii)...
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### Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
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### Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R.

Indicate True/False Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R. I need a hint to solve this problem. I have tried some common rings ...
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### Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
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### Polynomial ring with arbitrarily many variables in ZF

For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom ...
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### An example of a module that have no supplement.

We see that if R/J is not coclosed coprojective and J has a supplement then R/J is projective. Now we are looking for an example J has no supplement and also R/J is not coclosed coprojective. But we ...
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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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The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,... 2answers 44 views ### Given a ring$R$and a ring extension$R'$, if$r=r's$where$r'\in R'\setminus R$, does that mean that$s\not\mid r$in$R$? Given a ring$R$and a ring extension$R'$, if$r=r's$where$r'\in R'\setminus R$and$r,s\in R\setminus\{0\}$, does that mean that$s\not\mid r$in$R$? I was thinking for example in$\Bbb{Z}$, ... 2answers 89 views ### Is the ring$m\mathbb{Z}$isomorphic to the ring$n\mathbb{Z}$? I came over a question in ring theory which I am not being able to proceed upon: When is the ring$m\mathbb{Z}$isomorphic to the ring$n\mathbb{Z}$, where$m, n \in \mathbb{N}$? I know that to ... 1answer 104 views ### Prove that$\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$[duplicate] More precisely, given the ring homomorphism$\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with$\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where$\Bbb{R}[x,y]$is ... 1answer 23 views ### Is it true that the only regular elements in$Z_m\$ are invertible ones?

I have this doubt. In a unitary and commutative ring $$Z_m = \{[0]_m, [1]_m,\ ...\ ,\ [m - 1]_m\}$$ There are only two "kind" of elements: invertible and zero divisors. Is it true to say that the ...