For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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2
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0answers
49 views

Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
0
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0answers
21 views

Can one add reasonable assumptions that we have $depth\ R \geq depth\ M$ for every $R$-module $M$?

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. Here, Hanno shows that one can have any inequality between $depth\ R$ and $depth\ M.$ Still, the ...
1
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1answer
21 views

Example of module homomorphism from torsion to free

Can someone give an example of an R module homorphism from a torsion mkdule N to free module M where R is commutative with unity with the homomorphism different from the zero one I know if we dont ...
3
votes
2answers
76 views

Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
0
votes
0answers
8 views

Relationship between nonnegative real semiring module and boolean semiring module

In nonnegative matrix factorization, one attempts to factor a matrix $\mathbf{X} \in \mathbb{R}_{\geq 0}^{m \times n}$ into matrices $\mathbf{Z} \in \mathbb{R}_{\geq 0}^{m \times k}$ and $\mathbf{A} ...
2
votes
1answer
34 views

Find the structure of $\mathbb{Z}_{120}^*$

How to find the structure (in term of cyclic groups) of $\mathbb{Z}_{120}^*$? I know that the number of elements of $\mathbb{Z}_{120}^*= \phi(120) = 32 = 2^5$ But then, any hints?
1
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2answers
36 views

Proving module homomorphism has right inverse [duplicate]

I have met this as part of a problem in module theory which states Let $f : M \to U $ be a surjective module homomorphism over ring $R$ where $M$ is finitely generated and $U$ is free. How would I ...
2
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1answer
23 views

Proving $N/N'\cong$ Hom$_\mathbb Z(M'/M,\mathbb C^*)$

I am studying the following proof and need help understanding some of the details- Theorem : Let $N$ be a free $\mathbb Z$ - module of finite rank and let $N'$ have finite index in $N$ with ...
0
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1answer
44 views

Showing $ \mathbb Z/n\mathbb Z \oplus \mathbb Z/n\mathbb Z \ncong \mathbb Z/n^2 \mathbb Z $ as $\mathbb Z/n^2\mathbb Z$-modules

I'm trying to disprove that finite modules are isomorphic if and only if they have the same cardinality. The counterexample I'm trying to use is $$ \mathbb Z/n\mathbb Z \oplus \mathbb Z/n\mathbb Z ...
1
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1answer
12 views

$IM$ not finitely generated , $J \subseteq I$, $JM$ finitely generated; is there some $a\in I$ such that $JM\subsetneq\langle a,J\rangle M$?

Let $R$ be a commutative ring with unity, $M$ be an $R$-module, $I$ be an ideal of $R$ such that $IM$ is not a finitely generated submodule. Let $J \subseteq I$ be a finitely generated ideal such that ...
0
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1answer
11 views

Prove that $Ann_{R}(S)$ is a two-sided ideal of $R$ whenever $S$ is a submodule of $R$-module $M$

Let $M$ be an $R$-module and $S$ be a subset of $M$. The annihilator of $S$ in $R$ is $Ann_{R}(S) = \{ \lambda \in R; \forall x \in S \ \ \ \lambda x = 0 \}$. I need to show that $Ann_{R}(S)$ is an ...
0
votes
1answer
26 views

$M$ be a finitely generated $R$-module , and $N$ be a submodule of $M$ ; is it possible to have a meaning for $Ann(M)/N$ as an ideal?

Let $M$ be a finitely generated $R$-module, and $N$ be a submodule of $M$; is it possible to have a meaning for $Ann(M)/N$ as an ideal? (I ask this question due to its use in the third line in the ...
1
vote
1answer
26 views

Questions of a completely reducible module

Please help to deal with the tasks of: $1)$ Which cyclic groups are completely reducible as a $\mathbb Z$-modules? $2)$ Which cyclic modules are completely reducible over the ring $\mathbb F[x]$, ...
0
votes
1answer
29 views

a cyclic nonsimple module over $\mathbb{Z}$

Let $M$ be a module over $\mathbb{Z}$. Prove that: a.$M$ is a cyclic module if and only if it's a cyclic group. b.$M$ is simple if and only if it's cyclic of prime order. I tried ...
1
vote
1answer
22 views

Ring module homomorphism

Got peculiar question, not sure if my answer is correct. Let $R$ be ring and $b,r\in R$ Let also $f_b:R\rightarrow R$ where $f_b(r)=rb$ and $g_b(r)=br$ Why $f_b$ is R-module homomorpism, but $g_b$ ...
2
votes
1answer
11 views

A problem in determining conditions for relational matrix for module to be cyclic

I have met this question in my Abstract Algebra class dealing with modules, it is a problem I do not really understand how to do, which states: Let $ F $ be a field with characteristic 3 and we ...
2
votes
1answer
30 views

A question on finitely generated Abelian groups with a minimal number of generators

In my class on group theory I have encountered this strange looking question relating to Abelian groups in terms of generators which states: We are to find, up to isomorphism, all Abelian groups ...
0
votes
1answer
36 views

Module is semisimple if and only if all its finitely generated submodules are semisimple

I'm trying to show whether a module is semisimple if and only if all its finitely generated submodules are semisimple is true or not. I know that a semisimple module has semisimple submodules, ...
0
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3answers
43 views

Does any module with finite number of elements have a simple submodule? [closed]

Does any module with finite number of elements have a simple submodule? Not sure if this is true or not, struggling to find a counterexample
0
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3answers
74 views

Is $\mathbf{Q}/\mathbf{Z}$ semisimple as a $\mathbf{Z}$-module? [closed]

Is $\mathbf{Q}/\mathbf{Z}$ semisimple as a $\mathbf{Z}$-module? I feel as though this should be obviously not true, but struggling to show why.
1
vote
1answer
23 views

Elements of tensor product

Let $R, S$ be rings such that $R$ is a subring of $S$ and $1_R = 1_S$. Let $N$ be a left $R$-module. Let the free $\mathbb{Z}$-module on $S \times N$ be $F_\mathbb{Z}(S \times N)$. Let $H$ be the ...
1
vote
1answer
41 views

Prove that every $\mathbb{Z}/6\mathbb{Z}$-module is projective and injective. Find a $\mathbb{Z}/4\mathbb{Z}$-module that is neither.

I want to show that every $\mathbb{Z}/6\mathbb{Z}$-module is a direct sum of projective modules. I know that $\mathbb{Z}/6\mathbb{Z}$ is the direct sum of $\mathbb{Z}/2\mathbb{Z}$ and ...
0
votes
1answer
40 views

Can a subspace of a finite-dimensional vector space contain an infinitely ascending (or descending) chain of subgroups

If $V$ is a finite dimensional $K$-vector space, then every set of subsets of subspaces contains a maximal element, i.e. a subspace which no subspace of the set contains properly, equivalently we have ...
0
votes
1answer
17 views

Direct Sum and Intersection Query

I have been puzzling over a statement in Module Theory: Let $N\leq P\leq M$ (i.e. $N$ is a submodule of $P$ which is in turn a submodule of an $R$-module $M$.) Let $N'$ be another submodule of $M$. ...
1
vote
2answers
23 views

Tensor product with irreducible representation has no $G$-invariant submodules

Let $\rho: G \to GL(V)$ be a finite dimensional irreducible representation of a group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and let $R$ be a commutative ring with ...
0
votes
1answer
32 views

Proving that a module homomorphism is an isomorphism if and only if the induced map between the localizations is an isomorphism

Let $P$ be a prime ideal of a ring $R$. Let $M$ and $N$ be $R$-modules, and let $f:M \rightarrow N$ be a module homomorphism. Let $f_P: M_P \rightarrow N_P$ be defined by $f_P \left(\frac{m}{s} ...
3
votes
2answers
96 views

relations between homology and cohomology

Let $p$ be a prime number and $X$ a topological space. Are the following equivalent? (1) In the homology module $H_*(X;\mathbb{Z})$ there does not exist any element of order $p$. (2) In the ...
0
votes
1answer
38 views

Homology of Hom and Hom of homology

In 'Homological Algebra' by Cartan & Eilenberg: (page 203) For complexes $X$ and $Y$, consider the map $\alpha':H^{p+q}(\text{Hom}(X,Y))\rightarrow \text{Hom}(H_p(X),H^q(Y))$. Let $h_1\in ...
0
votes
2answers
80 views

If $M \otimes N \cong M' \otimes N$, is it true that $M \cong M'$? [duplicate]

I tried using the universal property of tensor products to show that there are mutually inverse maps from $M \times N$ to $M' \times N$, and use this to show that $M \cong M'$, but I didn't get far. I ...
0
votes
3answers
51 views

Showing that a ring homomorphism induces a correspondence that maps free modules to free modules

Let $f: \Lambda \rightarrow \Lambda'$ be a ring homomorphism. This induces a mapping $M \mapsto \Lambda' \otimes_{\Lambda} M$, where $M$ is a $\Lambda$-module and $\Lambda' \otimes_{\Lambda} M$ is a ...
2
votes
0answers
31 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow ...
1
vote
1answer
24 views

Number of submodules of $\mathbb{R}[X]/(X^2-a)$

Let $a\in\mathbb{R}$ and $M=\mathbb{R}[X]/(X^2-a)$ be a $\mathbb{R}[X]$-module. I want to prove: $M$ has exactly 2 submodules if $a$ is negative. $M$ has exactly 3 submodules if $a$ is ...
0
votes
1answer
11 views

No nontrivial cyclic modules imply module is simple?

I was reading a theorem: For a non-zero right $A$-module $N$, the following conditions are equivalent: (i) $N$ is simple (ii) $uA=N$ for all non-zero $u\in N$ I can understand (i) implies (ii), but ...
4
votes
2answers
47 views

Understanding of an example of “extending scalars”

The following is an example in the Abstract Algebra by Dummit and Foote: I don't understand in this example why $\iota$ is an isomorphism. By Theorem 8, I can get $$ id_N=\Phi\circ\iota $$ which ...
2
votes
2answers
24 views

Find the number of elements in each factorgroup.

(a) and (c) are 2 and 3 and I don't think I have a problem with those. However for (b) I get 11 cosets but they are not disjoint. According to theory there should only be 6. So what is happening ...
0
votes
1answer
35 views

Example of Artinian module which has infinitely many maximal submodules not isomorphic to each other

I'm looking for an Artinian module which has infinitely many maximal submodules not isomorphic to each other. I guess I can find it over a matrix ring.
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votes
1answer
35 views

Short exact sequence of $\mathbb{R}[X]$-modules that does not split [closed]

What is an example of a short exact sequence of $\mathbb{R}[X]$-modules that does not split?
3
votes
0answers
40 views

Homomorphisms of modules over a corner ring

Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules ...
4
votes
1answer
48 views

Basis for $\text{Mat}_2(\mathbb{Z})$ as a $\mathbb{Z}[i]$-module

Let $M=\text{Mat}_2(\mathbb{Z})$ a $\mathbb{Z}[i]$-module with scalar multiplication ...
4
votes
2answers
39 views

$A$ noetherian, $A$-endomorphism not injective for all invariant submodules is nilpotent

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module, $T: M \to M$ an endomorphism. Assume that for all $T$-invariant proper submodules $N$ of $M$, the induced endomorphism $\overline ...
1
vote
2answers
28 views

Checking if an element of tensor product is zero

I am trying to understand tensor product. Let $R$ be a ring and $R_0$ its subring. Then $R$ is a right $R_0$-module. Let $M$ be a left $R_0$-module. Is the element $1\otimes m$ of $R\otimes_{R_0} ...
1
vote
2answers
46 views

a property of the tensor product of modules

The following theorem is from the Abstract Algebra by Dummit and Foote (in the section 10.4 tensor products of modules): Would anybody illustrate how Theorem 8 is used to get $$ \textrm{ker ...
2
votes
1answer
59 views

$T$ automorphism of a finitely generated $A$-module, then $T^{-1} \in A[T]$

Let $M$ be a finitely generated $A$-module. Let $T: M \to M$ be an isomorphism. Then $T^{-1}$ is a polynomial in $T$ with coefficients in $A$. This seems to me as a direct application of ...
1
vote
1answer
66 views

Tensor products of simple modules over algebras

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and ...
2
votes
1answer
31 views

How do I show that $\text{End}_R\mathbb{Z}^n\cong\mathbb{Z}$ for $R=\text{Mat}_n(\mathbb{Z})$?

Let $R=\text{Mat}_n(\mathbb{Z})$ and $M=\mathbb{Z}^n$ the (left) $R$-module with action the matrix multiplication. How do I prove that $\text{End}_RM\cong\mathbb{Z}$? Should I find an explicit ...
2
votes
1answer
28 views

Proving that a $\mathbb{Z}[i]$-module is free

I have the $\mathbb{Z}[i]$-module $$A=\{(z,w)\in\mathbb{Z}[i]^2:z+(1+i)w=0\bmod(2+2i)\}.$$ I want to prove that it is free. I know that being free means that $A$ is generated by a linearly ...
1
vote
1answer
31 views

Functor right adjoint to $.\otimes_BA$

Given a ring morphism from $B$ to $A$, we can regard an $A$-module $M$ as a $B$-module. Then how can I prove the functor $._B:\operatorname{Mod}_A \to \operatorname{Mod}_B$ is right adjoint to ...
1
vote
1answer
18 views

What are the dual polyhedra of the face-centered cubic lattice?

For a given lattice $L$ we could define the set of points closest to one point more than any other. $$ \{ x \} = \min_{\ell \in L} \|x - \ell \| \in \mathbb{R}^3$$ This generalizes the "fractional ...
4
votes
2answers
58 views

Show that $a \otimes 1$ will vanish in $A\otimes_{\Bbb{Z}}\Bbb{Q}$ only when $a=0$.

Let $A$ be a $\mathbb{Z}$-module. I want to see that if $a \otimes 1=0$ in $A\otimes_{\Bbb{Z}}\Bbb{Q}$, then $a=0$. I'll assume $A$ is torsion free. Can I use that there exists an injective ...
0
votes
0answers
67 views

Decomposition of a graded module

In an article I am reading I found the following statement: If $D$ is a PID, then every finitely generated $D$-module is isomorphic to a direct sum of cyclic $D$-modules. That is, it decomposes ...