For questions about modules over rings, concerning either their properties in general or regarding specific cases.
1
vote
0answers
25 views
Indecomposable module over a PID, Jacobson's Basic Algebra I - followup
I am trying to solve the question which already appeared here:
Indecomposable $D$-module $M$, Jacobson's Basic Algebra I.
Let $D$ be a PID. Let $M$ be a finitely generated $D$-module. Then
$$
M\ ...
0
votes
1answer
26 views
Free modules and free submodules. [duplicate]
Just finished my semester in advanced abstract algebra and there was one question that I could not answer.
Let $R$ be commutative and let $(e_1,...,e_n)$ be a base for $R^{(n)}$. Put ...
2
votes
2answers
58 views
If $R[s], R[t]$ are finitely generated as $R$-modules, the so is $R[s + t]$.
Let $S \supset R$ as rings with $1 \in R$. Suppose that $s, t \in S$ and that the subrings $R[s], R[t]$ are finitely generated by $\{1, s, \dots, s^k\}$ and $\{1, t, \dots, t^m \}$. Then $R[s + ...
2
votes
2answers
29 views
Certain submodules of a finitely generated $R$-module are finitely generated?
Let $R \subset S$ be commutative rings. Let $s \in T$ where $T$ is a finitely generated $R$-module in $S$ with $R \subset T \subset S$. I want to show that $R[s]$ is finitely generated as an ...
0
votes
0answers
12 views
System of linear equations with modulo elements
Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the ...
0
votes
1answer
36 views
Sufficient condition for simple module
Every non-zero module homomorphism $f:M\rightarrow N$ is injective. Prove that $M$ is simple.
$f$ is not the zero map so $M$ is not $\{0\}$. I'm guessing I should let $N$ be a proper submodule of ...
-2
votes
0answers
35 views
Direct sum of tensor products of subalgebras [duplicate]
Let $R$ be an integral domain. Let $A,B$ two $R$-subalgebras of an $R$-algebra $C$, where $B=cA,c∈C$ and $K$ the field of fractions of $R$. If $(A\otimes_R K)+(B\otimes _RK)$ is a direct sum, is it ...
0
votes
1answer
18 views
For every positive integer $i$, a free module $F$ with basis $B_1$ over a ring $R$ with identity has a basis $B_{i+1}$ with $|B_{i+1}|=|B_1|+i+1$.
Let $F$ be a free module over a ring $R$ with identity. Suppose $F$ has two finite basis $B_1,B_2$ such that $|B_2|=|B_1|+1$. Prove that for every positive integer $i$, $F$ has a basis $B_{i+1}$ ...
0
votes
3answers
64 views
Question about tensor product of modules, when does $c \otimes r = 0$?
Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $C$ be an $R$-module. Then is $C \cong C \otimes_R R$ injectively contained in $C \otimes_R K$?
Define $\phi: C \otimes_R ...
-2
votes
0answers
64 views
Direct sum of tensor products [on hold]
Let $R$ be an integral domain. Let $A, B$ two $R$-subalgebras of an $R$-algebra $C$, where $B=cA, c \in C$ and $K$ the field of fractions of $R$. If $(A\otimes_R K) + (B\otimes_R K)$ is a direct sum, ...
1
vote
0answers
24 views
Modules over dedekind domain
Let $L$ be a finitely generated module over a dedekind ring $R$ and let $M$ be a submodule of $L$ such that $L/M$ is finite. I want to describe $L/M$. For a prime ideal $\mathfrak{p}$ of $R$, let ...
0
votes
0answers
28 views
Decomposing a module
I would like to decompose $M=\mathbb{Z}^2/\langle\mathbb{Z}(66,30) + \mathbb{Z}(12,4)\rangle$ as in the form given by the Main Theorem on Finitely Generated Modules.
I think $M$ is a ...
2
votes
1answer
18 views
Injective module over a quotient ring
I want to make sure I am not totally confused about this.
Suppose $R$ is an integral domain, $I$ a (non-zero) ideal of $R$. Then if $M$ is a (non-zero) $R/I$-module is it necessarily not injective ...
-1
votes
1answer
29 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.
2
votes
1answer
23 views
Classifying torsion-free injective modules over a PID
So the question is as in the title: Let $R$ be a PID, classify all torsion-free injective modules.
I know that it is going to be divisible, and using torsion free, if we define ...
0
votes
0answers
43 views
Show that we have an algebra homomrohpsim
I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field.
I suspect it's really easy but I don't know what to do. Is ...
1
vote
2answers
40 views
Tensor Product problem.
Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$.
I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$.
I know the second ...
0
votes
1answer
28 views
field extension $F(x)=F(x^2)$ [duplicate]
Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove?
I only obtained the proof for the ...
1
vote
4answers
73 views
Basis of Vector space $\Bbb C$ over rational numbers.
What will be the basis of vector space $\Bbb C$ over field of rational numbers?
I think it will be an infinite basis! I think it will be $B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$. ...
0
votes
1answer
47 views
Injective but not divisible module
I want to find an injective but not divisible $R$-module.
If $R$ is integral domain, every injective is divisible so it should be $R$ is not an integral domain. Is there any example?
1
vote
2answers
21 views
Why does the center of $R$-$\mathbf{mod}$ consist of multiplication maps induced from central elements of $R$?
The center of a category $C$ is the class of natural transformations from $1_C$ to $1_C$. In $R-\textbf{mod}$, I have been able to show that the morphisms $\eta_M(c):x\mapsto cx$ for $x\in M$ and $M$ ...
4
votes
3answers
80 views
similar matrices over $\mathbb{Z}$
Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
0
votes
0answers
36 views
composition series of modules
Let $R$ be a noetherian ring and $M$ an $R$-module of type $R/I$ for some ideal $I$ of $R$. Then we can define the composition series
$$0=M_0\subset M_1\subset \cdots\subset M_k=M$$
such that $M_i$ is ...
0
votes
1answer
34 views
matrices with same minimal and characteristic polynomials are conjugate
Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $A$, $B$ have the same minimal polynomials and characteristic polynomials, then can we prove that $A$, $B$ are conjugate? i.e., does ...
1
vote
1answer
37 views
Direct product versus direct sum [duplicate]
I want to know the rationale behind and the intuition of why they defined the tensor products. I started to read Keith Conrad's handouts. On page 4, paragraph 2 of this handout he says that the direct ...
1
vote
1answer
14 views
Span of a f.g. $R$-module over the quotient field of $R$.
Let $R$ be an integral domain with quotient field $K$, $R \neq K$. Let $V$ be a finite dimensional vector space over $K$ and $M$ a finitely generated $R$-module. My question is how is $K.M=V$ ...
0
votes
0answers
21 views
Can a presentation matrix present more than one abelian group?
Can a presentation matrix present more than one abelian group?
For example, $$
\begin{bmatrix}
2 \\
1 \\
\end{bmatrix}
$$
presents $\mathbb{Z}_2$. Since ...
2
votes
1answer
28 views
non degenerate bilinear map for modules
Let $R$ be a commutative ring with 1. Suppose $A,B$ are $R$-modules, $P:A\times B\to R$ is
a bilinear map that satisfies the following property: if $P(a,b)=0$ for all $b\in B$, then $a=0$. Then is the ...
1
vote
1answer
32 views
When is a simple tensor equivalent to natural multiplication?
Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules.
Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold?
$$x_1 \otimes x_2 = x_1x_2$$
The textbook I'm ...
2
votes
1answer
27 views
Flat modules on Stacks Project.
I have been reading through a bit of the material from the Stacks project, and there is a statement that I cannot make sense of. Lemma 10.36.19(7) states:
Let $R$ be a ring.
(7) Suppose ...
-1
votes
0answers
69 views
Why do we study modules? [closed]
Let $R$ be an associative ring. It is usual to study the category of $R$-modules and characterize some rings through treatments of some special modules. For example a ring $R$ is a noehterian if and ...
2
votes
3answers
57 views
$0$-th exterior power, empty product of modules and their tensor product
$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object
in the category of multilinear maps from $\finiteprod En$ where ...
1
vote
0answers
29 views
Vanishing criterion of pure wedges
Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
0
votes
1answer
100 views
RSA encryption/decryption scheme
I've been having trouble with RSA encryption and decryption schemes (and mods as well) so I would appreciate some help on this question: Find an e and d pair with e < 6 for the integer n = 91 so ...
1
vote
1answer
40 views
Every projective f. g. module is f. p.
I want to show that if $P$ is a projective f. g. module then $P$ is f. p.
0
votes
3answers
38 views
Free $\mathbb{Z}G$ module
Let $G$ denote a group and $\mathbb{Z}G$ the group ring. Let $F$ be a free $\mathbb{Z}G$ module over a set $X$. Why $F$ is free $\mathbb{Z}$ module over the set $\{gx|x \in X, g \in G\}$? Why is it ...
3
votes
1answer
86 views
A problem about an $R$-module that is both injective and projective.
Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$.
This is exercise 7.52 of Rotman's Advanced ...
0
votes
1answer
35 views
Construction of module on an Abelian group.
Let $M$ be an Abelian group and let $\operatorname{End} M$ be the set of all endomorphims on $M$. Then $(\operatorname{End} M,+,\circ)$ forms a unitary ring. In this setting, if $R$ is a unitary ring ...
0
votes
0answers
39 views
If $M$ satisfies ACC, do we have a similar result?
Let $M$ be an $R$-module and $f$ an $R$-endomorphism on $M$. If $M$ satisfies DCC then There exist $n\in\mathbb{N}$ such that $Imf^{n}+Kerf^{n}=M$.
If $M$ satisfies ACC, do we have a similar ...
4
votes
0answers
42 views
$M\times N$ Doesn’t Have a Module Structure
In Keith Conrad's notes (page 4) is written:
For two $R-$modules $M$ and $N$ , $M\oplus N$ and $M\times N$ are the
same sets, but $M\oplus N$ is an $R-$module and $M\times N$ doesn’t
have a ...
3
votes
1answer
48 views
Some questions about Fitting ideals
Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation
$$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$
we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ...
1
vote
3answers
79 views
Multiplicative inverse of 97 modulo 386?
A little help would be a lifesaver :)
I already used the Euclidean algorithm to find the GCD of $386$ and $97$, which is $1$. However, I'm stuck on this question posed by my professor: "Then use your ...
0
votes
1answer
40 views
Simple module and homomorphisms
Are the following statements equivalent?
i) $M$ is an $R$-simple module.
ii) Every $R$-homomorphism (nonzero) from $M$ to an $R$-module $N$ is a monomorphism.
iii) Every $R$-homomorphism (nonzero) ...
2
votes
1answer
44 views
How to determine the length of composition series of the $\mathbb{Z}$-module $\mathbb{Z}_{n}$?
How to determine the length of composition series of the $\mathbb{Z}$-module $\mathbb{Z}_{n}$?
Help me.
0
votes
1answer
41 views
Why we have $N_{1}+N_{2}+\cdots+N_{n}=N_{1}\oplus N_{2}\oplus\cdots\oplus N_{n}$?
Let $R$-module $M$ be a direct sum of modules $M_{1},M_{2},...,M_{n}$ and for each $i=\overline{1,n}$, $N_{i}\leq M_{i}$ ($N_{i}$ is submodule of $M_{i}$). Why we have ...
5
votes
1answer
69 views
On a commutative diagram
Let a commutative diagram:
\begin{array}{ccccccccc}
0 & \longrightarrow & A & \overset{f}{\longrightarrow} & B & \overset{g}{\longrightarrow} & C & \longrightarrow & ...
1
vote
1answer
52 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 $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?
2
votes
2answers
46 views
Can conclude that $N = f\left(K\right)\oplus f\left(L\right)$?
Let $K,L$ be two submodules of an $R$-module $M$ with the property that $K+L=M$ and $f:M\longrightarrow N$ is an $R$-epimorphism. Assume that $K \cap L = \ker f$. Can conclude that $N = ...
1
vote
1answer
19 views
$g$ is monomorphism and $M\cong N\oplus\left(M/g(N)\right)$
Let $M,N$ be two $R$-modules, $f\in Hom_{R}\left(M,N\right)$ and $g\in Hom_{R}\left(N,M\right)$ with the property that $f(M)=N$ and
$fg=Id_{N}$. Prove that $g$ is monomorphism and $M\cong ...
0
votes
0answers
22 views
Direct sum of submodule
Let $\left\{ M_{i}\right\} _{i\in I}$ be a family of submodules of
$R$-module $M$. Prove that $M$ is direct sum of $\left\{ M_{i}\right\}
_{i\in I}$ i.e $M=\oplus M_{i}$ iff for all $m\in M$, ...
