2
votes
1answer
20 views

Show that if $R$ is principal, $N $ is pure and $Ann(x+N)= Rd$ then there exists $y \in M$ such that $x+N=y+N$ and $Ann(y)=Rd$

Let $R$ a integral domain and $M$ a $R$-module. A submodule $N$ of $M$ is pure if for all $x \in M$ and $a \in R$ such that $ax \in N$, there exists $y \in N$ such that $ax=ay$. Show that if $R$ ...
6
votes
2answers
139 views

$M \oplus M \simeq N \oplus N$ then $M \simeq N.$

Let $M$ and $N$ be finitely generated $R$-modules where $R$ principal domain. Show that if $M \oplus M \simeq N \oplus N$ then $M \simeq N.$
0
votes
2answers
48 views

Show that an integral domain $R$ is principal if and only if every submodule of a cyclic $R$-module is cyclic.

Good morning, I have difficulty with this problem: Show that an integral domain $R$ is principal if and only if every submodule a cyclic $R$-module is also cyclic.
2
votes
2answers
80 views

Torsion-free but not free

The question is asking me to give an example of a finitely generated $R$-module that is torsion-free but not free. I remember in lecture, lecturer say something about the ideal $(2,X)$ in ...
0
votes
0answers
37 views

Find a counterexample to the following statement: $\frac{M_1 \oplus M_2}{N} \simeq M_2$

Suppose that $M$ is a $R$-module such that $M = M_1 \oplus M_2$ and let $N$ submodule if $M$ isomorphic the $M_1$. Find a counterexample to the following statement: $$\frac{M_1 \oplus M_2}{N} \simeq ...
1
vote
1answer
25 views

Show that if $M$ is a R-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$.

is it true that if $M$ is a $R$-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$. Comments: I tried to do the following: Suppose that $rm = ...
0
votes
1answer
35 views

Proves or counterexamples in retraction and coretractions of modules

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z} $ is not a ...
0
votes
1answer
37 views

An element annihilates a module

Let $R$ be a ring, $A$ an $R$-module, $y$ belong in $R$ such that $1+y$ annihilates $A$. Then for any ideal $I$ containing $y$, prove that $IA=A$.
0
votes
1answer
38 views

Direct product of rings and modules

Let $R_1$, $R_2$ be rings and $R=R_1\times R_2$. a) Let $M_i$ be an $R_i$-module for $i=1,2$. Show that $M = M_1\times M_2$ is an $R$-module where we define $(r_1,r_2)(m_1,m_2)=(r_1m_1,r_2m_2)$. b) ...
1
vote
0answers
54 views

Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free ...
0
votes
1answer
37 views

The sign representation of the Symmetric Group

I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. On page 11, after ...
0
votes
1answer
29 views

Constructing nontrivial $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
2
votes
0answers
72 views

If $\alpha$ and $\beta$ are algebraic integers then the roots of $x^2+\alpha x+\beta$ are algebraic integers

(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.) I'm trying to solve this problem with ...
0
votes
1answer
40 views

finitely generated right R-module which is not cyclic

Let $A$ be a finitely generated right $R$-module which is not cyclic. Prove that there exist $B \le A_R$ maximal with respect to the property that $A/B$ is not cyclic. help please and thank you for ...
0
votes
0answers
17 views

prove that if A is a finitely generated faithfully right $\Bbb R$-module

prove that if A is a finitely generated faithfully right $\Bbb R$-module ,then there exists $\Bbb { B\le A_R}$ such that $\Bbb {A/B} $ is faithful but $\Bbb {A/C}$ is unfaithful for all $\Bbb { ...
0
votes
0answers
51 views

Localization at prime ideal $(x_1, x_2, \dots)$ of infinitely many variables

I'm trying to solve a homework exercise which starts off: Let $k$ be a field, and let $A = k[x_1, x_2, \dots]$ be a polynomial ring in infinitely many variables. Let $\mathfrak p \subseteq A$ be ...
1
vote
1answer
44 views

Simple $R$-module

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true: 1) $N$ has a finite number of submodules. 2) $\operatorname{Hom}_R(N,N)$ is a division ring. 3) ...
0
votes
0answers
40 views

Module in $\mathbb{Z}$ and characteristics

I'd like to show some basic characteristics for a module in $\mathbb{Z}$. The module in Z has been defined as: "A subset $\not0 \not = M \subseteq \mathbb{Z}$ is called module, if M is closed under ...
0
votes
0answers
19 views

$R$ ring ,$ I,J$ sets and $I$ infinite. if $R^{(I)}$ izo of modules with $R^{(J)}$ then $| I | = | J |$.

I have to prove that. I know we have 2 cases. 1) If $| I | > | J |$ 1.1 $J$ infinite 1.2 $J$ finite 2) if $| I | < | J |$ this implies case 1.1 Please help me at case 1.2
2
votes
1answer
40 views

Localization of a module is zero implies the multiplicatively closed subset contains a single element annihilating the module

I need to show that if $S$ is a multiplicatively closed subset of a ring $A$, $M$ is a finitely generated $A$-module, and $S^{-1}M = 0$, then there exists a single element $s$ in $S$ so that $sM = ...
3
votes
1answer
49 views

The number of submodules of $M\oplus M$

Let $M$ is a simple $R$-module. Then what we can say about the number of submodules of $M\oplus M$? it can be infinite 2 3 4 I say that if $N$ be a submodule of $M\oplus M$, then for projective ...
2
votes
1answer
115 views

Annihilators and exact sequences

Let $R$ be a commutative ring. Let $M_1$, $M_2$ and $M_3$ be $R$-modules. Let the following sequence be exact: $$0\longrightarrow M_1 ...
0
votes
1answer
46 views

Surjective endomorphism of an $R$-module is injective.

I know this is a duplicate question. However, I haven't seen anything that invokes the isomorphism theorem. Here's my idea: By the isomorphism theorem we have that $M/\ker\varphi \cong ...
1
vote
3answers
105 views

Tensor products and monomorphisms of $A$-modules.

This problem addresses the same question that has been asked in $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$, which is exercise 2.11 of Atiyah and Macdonald's Introduction to ...
0
votes
3answers
67 views

Showing that simple modules over finitely generated commutative algebras are 1-dimensional and isomorphic to…

Let $A= \Bbb{C}[x_1, ... ,x_n]$. a) Show that every simple $A$-module is 1-dimensional and is isomorphic to $\Bbb{C}[x_1, ... , x_n]/(x_1 - a_1, ... , x_n - a_n)$ for some $a_1, ... , a_n \in ...
1
vote
2answers
102 views

If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
2
votes
2answers
94 views

Show that $Ker(f)$ is finitely generated, when $f: M \rightarrow A^n$ is a surjective A-module homomorphism and M is finitely generated

This is the problem I am attempting: Let $M$ be a finitely generated A-module and $f: M \rightarrow A^n$ a surjective homomorphism. Show that $Ker(f)$ is finitely generated. Following a hint in ...
2
votes
1answer
83 views

Prove that $J$ is an injective module.

Suppose $J$ is an $R$-module, and for every cyclic $R$-module $C$ this short exact sequence splits $$0\rightarrow J\overset{f}{\rightarrow}B\overset{g}{\rightarrow}C\rightarrow 0.$$ Prove that $J$ ...
0
votes
3answers
70 views

well defined mapping

Which of the following mappings is well-defined? a) $f: \mathbb{Z}_m \rightarrow \mathbb{Z}_m, \overline{x} \mapsto \overline{x^2}$ b) $g: {\mathbb{Z}}_m \rightarrow {\mathbb{Z}}_m, \overline{x} ...
1
vote
1answer
55 views

Proving a maximal element of a certain set of ideals is prime.

Let $R$ be a commutative ring and let $M$ be a nonzero $R$-module. If $m\in M$, define $\operatorname{ord}(m) = \{r \in R \mid rm = 0\}$, and define $F = \{\operatorname{ord}(m) \mid m \in M\ \wedge m ...
0
votes
0answers
47 views

Invariant factors of torsion module under homomorphism

Let M be a torsion module for a PID D with invariant factors $(d_1) \supseteq (d_2) \supseteq...\supseteq (d_r) $ (which means $M = Dz_1 \bigoplus Dz_2 \bigoplus ... \bigoplus Dz_r $ s.t $ann(z_i) ...
0
votes
0answers
26 views

Can a presentation matrix present more than one abelian group?

Can a presentation matrix present more than one abelian group? For example, $$ \begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} $$ presents $\mathbb{Z}_2$. Since ...
0
votes
1answer
106 views

If R/I is a free R-module, then I = 0.

Let I be an ideal of a ring R. Prove or disprove: If R/I is a free R-module, then I = 0. Need some hints. I know that a free R-module has a basis and is isomorphic to R$^n$ for some finite n. ...
0
votes
1answer
45 views

Show that $\mathrm{Ann}(M) = (8) \dots$, possible typo?

The question I was given is stated as follows: Let $M = \{1+a_1x+\dots a_7x^7 : a_i \in \{0,1 \} \} \subset \mathbb{Z}/2\mathbb{Z}[x]$. With group law $a(x)*b(x) = a(x)b(x) \mod x^8$, M becomes ...
0
votes
1answer
56 views

Prove equivalence: Internal Direct Sum of Modules (HW)

Let $M$ be a module and $M_1, M_2, M_3,..., M_n$ be submodules of $M$ verify that the following are equivalent: (1) $M$ is the internal direct sum of the $M_i$'s. (i.e. ...
1
vote
1answer
144 views

Projective Modules, Annihilators, and Idempotents

Let $Ra$ be the left ideal of a ring $R$ generated by an element $a \in R$. Show that $Ra$ is a projective left $R$-module if and only if the left annihilator of a, $\{r \in R | ra = 0\}$ is of the ...
0
votes
0answers
28 views

Set $S = \{(b_1, b_2, b_3, …) : b_i = \pm i! \mbox{ for all } i\}$

I'm attempting to do a problem involving free $\mathbb{Z}$-modules, and ran upon this description of a particular, well, I think it's supposed to be a $\mathbb{Z}$-module: $S = \{(b_1, b_2, b_3, ...) ...
0
votes
1answer
110 views

decomposition of cokernel

Lets work over $\mathbb Z$ and represent maps $f:\mathbb Z^l \longrightarrow \mathbb Z^n$ as matrices. For the following matrices write an equivalent diagonal matrix. I want to write a decomposition ...
3
votes
0answers
104 views

R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
0
votes
2answers
53 views

I'm not sure with one is correct…

At the class I wrote this at my notebook: $\frac{a}{d}\equiv\frac{b}{d}\pmod{n}\Leftrightarrow a\equiv b\pmod{n}$ Assume that $d\mid{a}\wedge d\mid{b}\wedge d\mid{n}.$ this is right? or It should be ...
2
votes
0answers
64 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
1
vote
1answer
65 views

Epimorphism from arbitrary module to a free module

Let $F$ be a free module and let $f:M \rightarrow F$ be an epimorphism from a module $M$ onto $F$. Show that there exists a homomorphism $h:F \rightarrow M$ such that $f \circ h = id_F$ and deduce ...
2
votes
1answer
407 views

Finitely generated modules over a Noetherian ring are Noetherian

I'm trying to prove that if the ring $R$ is Noetherian then every finitely generated $R$-module is Noetherian. First of all, it is known that every module is a homomorphic image of a free module, ...
1
vote
1answer
41 views

Morphisms of complexes chain [closed]

I have a small question: Why is the following true? "If we have a continuous mapping between two topological spaces $f:X\rightarrow Y$, we can associate a morphism of chain complexes $f_*\colon ...
-1
votes
1answer
90 views

Module, vector space and abelian group

please can someone help me to prove this two properties $\bullet$ If K is a field, then the concept of K-vector space (a vector space over K) and K-module are identical. $\bullet$ The concept of a ...
2
votes
0answers
58 views

Structure Theorem for finitely generated modules over PIDs [duplicate]

Let $A$ be a real $4 \times 4$ matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either $$\begin{pmatrix} ...
4
votes
1answer
62 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
1
vote
1answer
76 views

Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
3
votes
0answers
70 views

Help understanding a comment about isomorphisms of two direct sums.

I've recently noticed a few different posts asking about this problem Isomorphism of two direct sums, all using the CRT. I think I understand it this way. What caught my eye is in the above link a ...
2
votes
1answer
128 views

Modules over Principal Ideal Domains

Let $a$ and $b$ be two nonzero elements of a PID $R$. Prove that direct sum $R/(a)\oplus R/(b)$ is isomorphic to the direct sum $R/(u)\oplus R/(v)$, where $u=\gcd(a,b)$ and ...