# 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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### Please help in proving $ab=0 \space \forall a,b$ in a ring $R$ where the only right ideals of $R$ are the trivial ones and $R$ is not a division ring.

I have the following question I am trying to solve: Let $R$ be a ring such that the only right ideals of $R$ are $(0)$ and $R$. Prove that either $R$ is a division ring or that $R$ is a ring with ...
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### Rings with bounded index of nilpotency are Dedekind-Finite

Recently in an article by A. A. Klein I have seen this result: A ring $R$ with bounded index of nilpotence is Dedekind-Finite. Can anyone help me prove this result?
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### Counting rings of order $p^3$

MathWorld states that there are exactly 52 rings of order 8 (multiplication in rings may be not commutative and perhaps there will be no neutral element) and 53 rings of order $p^3$ where $p$ is an ...
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### finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
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### Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...
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### To prove ; $pa:=a+a+… p$ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ for some prime $p$ then the ring $R$ is commutative

If in a ring $R$ , $\exists$ prime $p$ such that $pa:=a+a+... p$ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ , then how to prove that $R$ is commutative ? I would not want to use ...
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### Example of finite ring which is not a Bézout ring

A left (or right) Bézout ring is a ring in which any two elements generate a principal left (resp. right) ideal. Assume that we have a finite ring R. Does there exist some classification theorem (...
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### Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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### Counterexamples to the Artin-Rees Lemma

This well known Lemma about $I$-stable filtrations asserts: Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module. Let $F$ be a submodule of $E$ and $\{E_i\}$ an $I$-...
### $G \simeq R^{\times}$
What is known about the groups G for wich there exist a unitary ring R, such that $R^{\times} \simeq G$? I can easily prove that The only G cyclic with this property(Edit:and odd order) are those who ...