# Tagged Questions

44 views

### Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
37 views

### A finite $\mathbb{Z}$-module whose submodules are totally ordered by inclusion.

I have the following problem: Let $M$ be a finite $\mathbb{Z}$-module such that set of the submodules is totally ordered by inclusion. Prove that there exist a prime $p$ such that $|M|=p^\alpha$ ...
30 views

### A proposition on exact sequence of inverse limit (Lang, Algebra, p. 165)

I am trying to understand this proof. My only question is that what are the vertical maps here?
53 views

### Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
29 views

### Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
334 views

### If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
108 views

145 views

### Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
123 views

### Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence $$0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0.$$ ...
68 views

### Why $\mathbb Z_{p^{\infty}}$ is an artinian $\mathbb Z$-module? [duplicate]

If $M=\{\frac{a}{p^{n}}| a\in\mathbb Z , n\in\mathbb N\}$ and quotient group $(M/\mathbb Z,+)=\mathbb Z_{p^{\infty}}$. Prove that $\mathbb Z_{p^{\infty}}$ is an artinian $\mathbb Z$-module.
61 views

### Is $\mathbb{Z}$ isomorphic to a direct subproduct of the family $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?

Is $\mathbb{Z}$ isomorphic to a direct subproduct of the family $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?
167 views

### $p$ prime, $P = \left\{ \frac{m}{p^e} \middle| m, e\in \mathbb{Z} \right\}$. Prove that $\mbox{Ext}(P; \mathbb{Z}) \cong \mathbb{Z}^{(p)}/\mathbb{Z}$

I don't know why the book Homology by Saunders Mac Lane is wwaaayyy tttoooo hard to digest. :((( This is like the third time I read this book, but still not clear is everything, and to tell the ...
61 views

### Modules decomposition into indecomposables

I think it's not true that every module (over arbitrary ring) is the sum of indecomposable modules, but I can't find counterexample and literature about this problem. Can anyone help me? Also I have ...
368 views

### Finding invariant factors of finitely generated Abelian group

There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe). Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...
102 views

### Intersection of direct summands of torsion free abelian groups

Suppose that $G_1$ and $G_2$ are direct summands of torsion free abelian group $G$. Must $G_1 \cap G_2$ also be a direct summand? It is true when $G$ is free, at least.
82 views

### $G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]

Possible Duplicate: Isomorphism between $I_G/I_G^2$ and $G/G'$ Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal of the integral group ring $\mathbb{Z}[G]$. ...
148 views

159 views

### Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
153 views

### Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
298 views

### Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$

Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$ Since $\mathbb{C}[x]$ s noetherian we have that $\mathbb{C}[x]/(x^2)$ is too. And thus finitely generated ...
108 views

### Finding subgroups of index 2 of $G = \prod\limits_{i=1}^\infty \mathbb{Z}_n$

I looked at this question and its answer. The answer uses the fact that every vector space has a basis, so there are uncountable subgroups of index 2 if $n=p$ where $p$ is prime. Are there ...
233 views

### Torsion module and its socle

A torsion abelian group has nonzero socle and is an essential extension of it. Let $R$ be a commutative ring. If $M$ is a unital, torsion $R$-module with nonzero socle, is $M$ an essential extension ...
93 views