# 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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### 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 ...
• 348
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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 ...
38 views

### 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 ...
• 125
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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 ...
• 2,937
1 vote
43 views

### Existence of one finite presentation implies all other presentations are also finite for modules?

Given a finite presentation $A^m\to A^n\to M\to 0$ of M, does it mean for any other presentation $\text{Ker}(f)\to A^k\xrightarrow[]{f} M\to 0$, the relations $\text{Ker}(f)$ is also finitely ...
• 1,958
1 vote
39 views

### Is the intersection of finite submodules a finite submodule? [duplicate]

If $M$ is an $A$-module and $N,P$ are finite(ly generated) submodules of $M$, is is true that $N\cap P$ is also finite? I cannot think of a counterexample right now, but I neither see how one would ...
1 vote
52 views

### Minimum size of a minimal generating set of a finite abelian group

I am trying to prove that the minimum size of a minimal generating set of a finite abelian group $G$, denoted $d(G)$, where $G=C_{d_{1}} \times \dots \times C_{d_{k}}$ for $d_{i} \mid d_{i+1}$, is $k$....
• 11
1 vote
21 views

### Noether Normalization and Lie Algebra of derivations on a commutative algebra $A$

In my Lie algebra course, the professor asked us to use Noether's Normalization Theorem and use it to say something about the Lie algebra of derivations over a commutative, associative algebra. Now, ...
• 5,518
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### Finitely generated abelian groups $G,H$ with the same finite quotients (up to isomorphism) are isomorphic

Two finitely generated abelian groups $G,H$ have the same finite quotients (up to isomorphism). Prove they are isomorphic. According to an earlier result, it seems that I need to use the fact that ...
• 1,555
20 views

### Unitary Character Group: Book request

I am looking for books discussing unirary character groups. In particular, I am interested in the case where $G$ is a finitely generated abelian group. I suppose different authors use different ...
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### Dimension and length

I was trying to prove the following: Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module. Then $M$ has finite length if and only if $\mathrm{dim}(M)=0$. I was able to prove ...
• 121
107 views

### A module over PID which is not direct sum of cyclic mdules [duplicate]

I am trying to solve the following qualifying exam problem: “Give an example of a module over a PID that is not isomorphic to a direct sum of cyclic modules. Justify your example”. (Carnegie Mellon, ...
• 401
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### Smallest possible cardinality of the generating set of a group

A very useful invariant associated to any group is the smallest possible cardinality of a generating set that generates that group. This is always guaranteed to exist, no matter what the group, at ...
• 7,082
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### Showing $2 Q = Q$

I want to show these two statements: 1- Show that $2 \mathbb Q = \mathbb Q.$ 2-Show that $\mathbb Q$ is not a finitely generated $A$ module. For the first statement I do not know how to show it, so if ...
48 views

### $M$ projective and finitely generated implies $Hom_R(M,R)$ is projective and finitely generated

I'm trying to prove that if $M$ is a left $R$-module projective and finitely generated, then $Hom_R(M,R)$ is a right $R$-module projective and finitely generated. I've seen comments where they use ...
• 51
1 vote
28 views

### A field that is a finitely generated $k$ algebra is also a finitely generated $k$-module

Let $F$ be a field, let $k$ be a subfield of $F$, then $F$ is a finitely generated $k$-algebra iff it's a finitely generated $k$-module. The $\impliedby$ part is obviuos, so I'm only interested in the ...
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### If localizations at generators of $A$ are f.g $B$-algebras so is A

Let $A$ be a ring, $A=(a_1,\dots,a_n)$ with $\sum_{j=1}^nx_ja_j=1$ for some $x_1,\dots,x_n\in A$ and suppose that $A_{a_k}$ are finitely generated $B$-algebras for some ring $B$. The end goal is to ...
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Consider a fixed infinite group, such as $SL(n, \mathbb{R})$. Let $S$ be a given finite subset of the elements of $SL(n, \mathbb{R})$, further suppose that $S$ is closed under inverses. Is there any ...