Questions tagged [ring-theory]

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

14,496 questions
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UFD that is not a ED?

Can someone give some examples of a unique factorization domain, that is not a Euclidean domain? I'm aware of $\mathbb{Z}[\frac{1}{2}+\sqrt{-19}]$ and would appreciate any other examples, the simpler ...
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Find a group epimorphism $\mathbb{Z}_m \rightarrow \mathbb{Z}_{(m,n)}$ with kernel $n\mathbb{Z}_m$

Let $m$ and $n$ be positive integers and $\otimes=\otimes_\mathbb{Z}$. Denote the GCD of $m$ and $n$ by $(m, n)$. I proved $\mathbb{Z}_m \otimes \mathbb{Z}_n \cong \mathbb{Z}_{(m,n)}$ by considering ...
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$(0)$ and $p^n\mathbb{Z}$ (where $p$ prime, $n$ positive integer) are the only primary ideals in $\mathbb{Z}$

I am trying to show that $(0)$ and $p^n \mathbb{Z}$ are the precisely primary ideals in $\mathbb{Z}.$ Clearly $(0)$ is a prime ideal hence primary and radical of $p^n \mathbb{Z}$ being the maximal ...
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If $\mathbb Z_{15} / P \cong \mathbb Z_{3}$, then prove that $\mathbb Z{_{15}}_{P} \cong \mathbb Z_{3}$

Here, $\mathbb Z{_{15}}_{P}$ is the localization of the integers by the prime ideal $P$, that is, $\mathbb Z{_{15}}_{P}$=$D^{-1}Z{_{15}}$, for $D = \mathbb Z{_{15}}-P$. I tried this problem by brute ...
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If $R[x]$ is a PID, $R$ necessarily has to be a field?

It is given that R is a commutative ring with identity. My attempt: I tried to get a contradiction. Given a nonunit $a \in R$, I wanted to show that $(a,x)$ is not a principal ideal in $R[x]$. Then, ...
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$A$ is a field iff $A[t]$ is euclidean.

I'm almost sure the question has already been asked but i don't know the english terminologies... I have in my lecture that : $A$ a ring. $A$ is a field iff $A[t]$ is principal. I'm anoyed ...
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Prove $F[x]/(x^n)$ is an injective module

Let $F$ be a field and $n\geq1$ (1) Prove $R=F[x]/(x^n)$ is an injective $R$-module. (2) Give a projective resolution and an injective resolution of the $R$-submodule $M=(x)/(x^n)$ For part (1), I ...
In $\mathbb Z[x]$, is $(2,x)=(2)+(x)$?
The text says that $(2,x)=(2)+(x)$, because $1 \in \mathbb Z$. I do not see why this leads to the decomposition. Can someone point me in the right direction?