# Questions tagged [finitely-generated]

For questions regarding finitely generated groups, modules, and other algebraic structures. A structure is called finitely generated if there exists a finite subset that generates it.

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### Is every unitary ring finitely generated?

I'm puzzled by the following: if $R$ is a unitary ring then $R$ is generated by $1_R$, denoted as $$R = \langle 1_R \rangle.$$ can we conclude that every unitary ring is finitely generated? I know ...
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### On which rings must a finitely generated module be finitely presented? Is there an 'if and only if' characterization for such rings?

As is well known, if $R$ is a Noetherian ring, then a finitely generated module over $R$ must be finitely presented. However, this is not necessarily true for coherent rings. For example, consider $k$ ...
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### Are the following definitions of a finitely generated $k$-algebra equivalent?

In my lecture notes for algebraic geometry, an algebra over a field $k$ is defined as a (unital and commutative) ring, together with a ring homomorphism $\lambda:k\to R$ (such a homomorphism preserves ...
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### Show that $(\Gamma : 2\Gamma)<\infty$ for abelian $\Gamma$ implies that $P\mapsto \frac12P$ cannot go on forever

I was recently reading this question on the relation between Mordell's Theorem and FLT for the $n=4$ case. Reading more carefully Knapp's book, and more precisely the chapter where he discusses the ...
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### Is it possible for a normal subgroup of a finite group have greater number of elements in the minimal generating set?

Let $G$ be a finite group, and $1 \lhd N \lhd G$. With $G = \langle A \rangle$ and $N = \langle B \rangle$ be minimal. Is it possible for $|B|>|A|$? Main motivation behind this question was ...
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### linear isoperimetric inequality implies hyperbolicity

I am trying to find a nice proof that a finitely presented group satisfying a linear isoperimetric inequality implies it is hyperbolic. I came across these lecture notes, Theorem 3.22, but I am having ...
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### Linearly Compact Module in $R-Mod$

Definition: A module $M$ is called linearly compact if for a family of cosets $\{x_{i}+M_{i}\}_{\triangle}$, $x_{i}\in M$, $\triangle$ is a directed set, and submodules $M_{i}\subset M$ (with $M/M_{i}$...
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### If $R$ is a Noetherian ring, then $R^n$ is Noetherian [closed]

I'm working with a Noetherian ring $R$. As an $R$-module, $R^n = R \oplus ... \oplus R$. I want to show that $R^n$ is Noetherian in the sense that it obeys the ascending chain condition for its ...
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### optimization problem over finite groups (or at least finitely generated groups)

Have you ever seen any optimization problem over finite groups (or finitely generated groups)? That is, given a group $G$, we want to maximize or minimize a function $f$ over $G$. An example that ...
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### If an Ideal I is not finitely generated and then I+(a) is not finitely generated.

Suppose of have a ideal $I\subset R$ that is not finitely generated. Then is it the case that the ideal $I+(a)$ is also not finitely generated. I was thinking to assume the contradiction that it is ...
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### What is the order of a group given three generators, $a^2=b^3=c^4=1$ and $cb=ac$
A group $G=\langle a,b,c\mid a^2=b^3=c^4=1, cbc^{-1} = a\rangle$ what is the order of the group $G$ give all such possible values. My attempt: Since $cbc^{-1} =a \Rightarrow cb = ac\;\;(*)$, but then ...