Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
0
votes
0answers
17 views
Socle is simple.
Let $M$ be a module with uniform dimention 2 and socle of $M$ is simple. Assmume $X$ and $Y$ are two uniform submodules of $M$ such that $X+Y$ is essential in $M$. Can we say $X\cap Y=0$?
0
votes
0answers
38 views
how to prove that $\Bbb Q$ is a module of $\Bbb Z$ with no basis? [duplicate]
how can it be proven that $\Bbb Q$ is a module of $\Bbb Z$ with no basis?
0
votes
0answers
10 views
An isomorphism of modules and an action of a Galois group
Let $K \subset L$ be a finite Galois extension and $\text{Gal}(L/K) = G$. Suppose we have a vector space $V$ over $L$ and an isomorphism $\alpha: V \otimes_K L \to V \otimes_K L$ between $L \otimes_K ...
0
votes
0answers
17 views
Positive solution of a linear system with integer coefficients
Suppose that $b_1,\dots,b_m \in \mathbb Z^n$ are not linearly independent over $\mathbb Z$ (otherwise the problem is trivial). Given another element $w\in \mathbb Z^n$ is there a way to determine (...
4
votes
1answer
44 views
About the definition of coinduction of a module
If $G$ is a group, $H$ a subgroup, and $N$ a left $\mathbb{Z}[H]-$module, I've learned the following construction:
$$\mathrm{coInd}_H^G(N) = \mathrm{Hom}_{\mathbb{Z}[H]}(\mathbb{Z}[G], N)$$
where $\...
1
vote
1answer
8 views
Reduced rings and simple modules
I'm having trouble getting my intuition pumping and the details in my head on the Jacobson radical (intersection of maximal ideals or intersection of annihilators of simple modules). What I'm ...
1
vote
1answer
30 views
In R-Mod Category, example for $B\cong A \oplus C \not\implies 0 \to A \to B \to C \to0$ splits.
https://en.wikipedia.org/wiki/Splitting_lemma
In $R$-Mod Category, short exact sequence
$0 \to A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \to0$ splits
if it satisfies one of the ...
0
votes
0answers
22 views
Interpretation of the cardinality of cokernel
I am wondering the interpretation of $|\operatorname{coker}\phi|$, where
$$\phi:\mathbb{Z}[t]^n\rightarrow\mathbb{Z}[t]^n,(p_1(t),...,p_n(t))\mapsto(p_1(t),...,p_n(t))\cdot A,$$
and $A$ is an $n\...
1
vote
0answers
20 views
Does $\DeclareMathOperator{\len}{length}\DeclareMathOperator{\rk}{rank}\len(M/xM) \leq \rk(M) \cdot \len(R/(x))$ hold over non-integral rings $R$?
$\DeclareMathOperator{\len}{length}
\DeclareMathOperator{\rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says:
The basic result of this section expresses ...
-1
votes
0answers
25 views
About irreducible representations over the polynomial ring $k[x]$
From Example 2.3.14 (2) here page 21: Let $A = k[x]$. Since this algebra is commutative, the irreducible
representations of $A$ are its 1-dimensional representations. They are defined by a single ...
3
votes
1answer
44 views
Is $R$ an algebra or not?
In the book Quasi Frobenius Rinngs- Nicholson and Yousif, Example 2.5 gives a ring as follows:
Let F be a field and assume that $a→ \bar{a}$ is an
isomorphism $F → \bar{F} ⊆ F$, where the subfield $\...
0
votes
2answers
24 views
How to take an absolute value or modulus of z?
let's assume the :
$$z=\frac{e^{(-jc)}}{(a+jb)}$$
I would like to take the absolute value of z.
I started with multiplication z with $\frac{(a-jb)}{(a-jb)}$ and got:
$$abs\frac{e^{(-jc)}}{(a+jb)}=\...
0
votes
0answers
18 views
Invariant basis number for Rings.
Let $R$ be a ring with identity. Let $M_{n}(R)$ be the ring of $n$ by $n$ matrices with entries in $R$. Prove that $R$ has IBN iff $M_{n}(R)$ has IBN.
I was given this problem. I thought a little ...
0
votes
1answer
23 views
prove the next algebras isomorphism
Let V be a non trivial vector space s.t. $dim V = n$ over $\mathbb{C}$. also let $T:V\to V$ be a linear transformation over $\mathbb{C}$.
now we shall define A = $\mathbb{C} [x]$ to be the ...
0
votes
0answers
23 views
Calculation of Tor for module with trivial action
Let $S$ be a ring of the form $\mathbb{Z}/p[x] \otimes_{\mathbb{Z}/p}E(y)$. Where $\mathbb{Z}/p[x]$ is the polynomial algebra and $E(y)$ is the exterior algebra over $\mathbb{Z}/p$. Consider $\bar{S},...
1
vote
1answer
22 views
What do “projections” out of tensor products look like?
Convention. If $R$ is a relation $X \rightarrow Y$, what I mean is that $R$ is a subsets of $X \times Y$. We say that $X$ is the domain of $R$ and that $Y$ is the codomain. We'll write $x \overset{R}{\...
2
votes
3answers
71 views
What do brackets mean for mod operation?
I'm solving equation 5 = (6 * 8 + 9 * b)(mod 10). I tried to use wolframalpha and it gives me answer b = 3. But if I remove ...
2
votes
2answers
24 views
Surjective module homomorphism? $0$ module homomorphism?
I am trying to resolve an exercise and there are 2 point that are missing in order to finalize:
Suppose $A$, $B$, $C$, and $P$ are $R$-modules, and $f:A \rightarrow B$ and $g:B\rightarrow C$ are both ...
0
votes
1answer
14 views
If possible, construct a simple module that is not projective.
Suppose that $M$ is a simple (or irreducible) $R$-module. Does this imply that
$M$ is projective?
Vector spaces are all projective since they are free, so no counter example can be gotten from there....
2
votes
1answer
32 views
If $I$ is an ideal of a ring $R$, then for any $a\in R$, is there a difference between $(R/I)a$ and $Ra/I$?
If $I$ is an ideal of a ring $R$, then for any $a\in R$, is there a difference between $(R/I)a$ and $Ra/I$?
I am seeing the following:
$$(R/I)a=\{r+I: r\in R\}a=\{ra+I:r\in R\}=Ra/I.$$ Does this ...
-2
votes
1answer
67 views
Proving that $\operatorname{Hom}(\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z},$
Proving that $\operatorname{Hom}(\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}$ using short exact sequences.
A short exact sequence $0 \to A \to B \to C \to 0$ ...
2
votes
1answer
24 views
If the completion of a module is trivial, must the module be trivial?
I want to prove the following lemma as a step in solving an exercise from Atiyah-MacDonald's Commutative Algebra.
Claim: Let $A$ be a Noetherian ring, and $M$ a finitely generated $A$-module. Let $I \...
2
votes
1answer
25 views
Extensions of indecomposable modules
Let $R$ be a unital ring. Suppose that $A$, $B$, and $C$ are unitary left $R$-modules such that there exists a non-split exact sequence
$$0\to A \overset{\alpha}{\longrightarrow}B\overset{\beta}{\...
0
votes
0answers
26 views
View ${\operatorname{Hom}_A(F(B),M)}$ as subset of $\operatorname{Hom}_k(F(B),M)$?
Given a finite dimensional $k$-algebra $A$ and two (finitely generated) left $A$-modules $M, N$ we can view $\text{Hom}_A(M,N)$ as a subset of the k-linear maps $\text{Hom}_k (M,N)$.
Now also ...
0
votes
0answers
12 views
Is the product of two Smith Normal Forms a the Smith normal form of the product?
Suppose A and B are square matrices of the same size over a PID R. Does the Smith Normal form of AB equal the product of the Smith normal form of A and B?
I think this should be false. However, I can'...
1
vote
1answer
44 views
Noetherian domain whose fraction field is such that some specific proper submodules are projective
Let $R$ be a Noetherian domain (which is not a field) with fraction field $K$. Suppose every proper $R$-submodule of $K$ of the form $R[1/a]$, where $a\in R$, is projective as an $R$-module. Then, I ...
0
votes
0answers
64 views
How do you write this module as a direct sum of cyclic modules?
Let M be the module over Z [i] generated by elements x, y whose relations are determined by $(1+i)x+(2-i)y=0$ and $3x+5y=0$. How can one write M as a direct sum of cyclic modules?
1
vote
1answer
53 views
$I$-torsion functor and $\sqrt{I}$
Let $R$ be a Noetherian ring and $I,J\subseteq R$ ideals. For any $R$-module $M$, define $\Gamma_I(M)=\{x\in M\mid I^n x=0\text{ for some }n\in\mathbb{N}\}$. Suppose that $\Gamma_I=\Gamma_J$. I want ...
0
votes
0answers
48 views
$M$ is free $R$-module $\iff$ $M$ has $R$-basis
We will define the free $R$-modules.
Definition. Let $R$ be a ring with $1_R$ and $F$ an left $R$-module. We call $F$ free $R$-module, if $$F=\bigoplus_{i\in I}
R_i$$ where $R_i:=\langle b_i \...
3
votes
1answer
48 views
diagonal and codiagonal morphisms in additive category
Consider $R$ a commutative ring and $M$ a $R$-module.
I know that the diagonal homomorphism is defined as follows:
$\Delta_M : M \longrightarrow M \oplus M : m \longrightarrow (m,m)$
I also know ...
0
votes
1answer
69 views
equality of two simple tensors in $R=k[x,y]$
Consider $R=k[x,y]$, where $k$ is a field. Consider the $R$-module $M=\langle x,y\rangle$.
I would like to see that $x \otimes y \neq y \otimes x$ in $M \otimes_R M$.
My try to prove it:
Let $F$ ...
0
votes
0answers
9 views
criteria for a R-module to be monogenous
M is a finitely generated R-module where R is a principal ring.
Let $M = Am_1+....+Am_r$ where $M=<m_1,...,m_r>$.
let $p=GCD(m_1,...,m_r)$.
A is principle so $M=Ap$.
Does it mean that every ...
0
votes
1answer
28 views
Z-module isomorphism
Let n,d be positive integer numbers such that d|n. Show that $<\frac{n}{d}>$/$<n>$ Is isomorphic as a module to $\mathbb{Z}_{d}$
0
votes
1answer
51 views
Projective cover of modules
Give me a hand please, how can i prove this sentence.
"Show $\mathbb{Z}_{2}$ doesn't have projective cover as $\mathbb{Z}$-module."
1
vote
1answer
45 views
About Loewy length of syzygy
Suppose $\Lambda$ is an Artin algebra with $\DeclareMathOperator{\rad}{rad}\rad^3\Lambda=0 $, and $ M $ any finite $\Lambda$-module with projective dimension finite. Proof that $\Omega M $ has Loewy ...
2
votes
1answer
28 views
Question about $\text{Hom}_{\mathfrak{g}}(M,L)$
Let $\mathfrak{g}$ be a complex semisimple Lie algebra.
Suppose $M,N,L$ are $\mathfrak{g}$-modules, $N$ is a $\mathfrak{g}$-submodule of $M$.
Does this implies $\text{Hom}_{\mathfrak{g}}(N,L)\le \...
1
vote
2answers
27 views
Clarification on this submodule over $\mathbb{Z}[t]$
I am having trouble making the following calculation consistent. Consider the map from a $\mathbb{Z}[t]$-module to itself:
$$\phi:\mathbb{Z}[t]/I\rightarrow \mathbb{Z}[t]/I,\\p(t)+I\mapsto (p(t)+I)(t^...
0
votes
1answer
26 views
Alternative definition on free modules.
I 'm trying to compare Free Modules and Free Abelian Groups.
We know that,
Definition. An abelian group $G$ is called free abelian group with rank $n\in \Bbb N^*$, if $G$ is the direct sum of $n$ ...
0
votes
1answer
26 views
Can I Find A Map from a Module M to the kernel of a map p from M to M?
I have a module homomorphism $p:M\rightarrow M$. I would like to find another module homomorphism $\phi:M\rightarrow \ker(p)$. Finding such a thing seems to be very challenging however. Is this ...
0
votes
0answers
20 views
Modulo arithmetic all square pairs
Let $n, m\in\mathbb{N}, n\neq0$. Find the number of all pairs $(i,j)$ such that $(i^2+j^2) \bmod m =0, i,j=1..n$
I want to do this in the least amount of steps.. All I can think about is that $(a+b)\...
4
votes
3answers
46 views
Exact sequence with finite groups: $0 \to A \xrightarrow[]{\alpha} \mathbb{Z}^d \xrightarrow[]{\beta} \mathbb{Z} \xrightarrow[]{\gamma} B \to0$.
I have this exact sequence of abelian groups:
$$0 \to A \xrightarrow[]{\alpha} \mathbb{Z}^d \xrightarrow[]{\beta} \mathbb{Z} \xrightarrow[]{\gamma} B \to0$$
with $A$ and $B$ finite abelian groups and $...
1
vote
1answer
35 views
Finitely generated free module is projective.
Call a $R$-module projective if every short exact sequence $0 \to A\stackrel{f} \to B\stackrel{g} \to C \to 0$ of $R$-modules splits.
Call a short exact sequence as above split, if it admits a ...
0
votes
0answers
22 views
Express the following module as a direct sum of cyclic R-modules
Now
$\dfrac{R}{\langle (x-x^2)\rangle }\cong \dfrac{R}{\langle (x)\rangle }\oplus \dfrac{R}{\langle (1-x)\rangle }$ since $x,1-x$ have no common factors so by Chinese Remainder Theorem we can say ...
0
votes
1answer
23 views
What module is of finite length whose submodules aren't? [closed]
Does such an example exist? It seems strange that a module whose submodules have infinite length has somehow infinite length
1
vote
1answer
28 views
On showing that $\varphi(N)=(N+K)/K$, for modules $N,K<M$, $\varphi$ natural map.
In this question modules are A-modules, where A is a unitary commutative ring
This seems trivial, but beginning with module theory, I was trying to see why $\varphi(N)=(N+K)/K$, for the natural A-...
6
votes
2answers
82 views
If $A\cong B$, then $A\otimes C\cong B\otimes C$.
I think this is kind of true, since $\square\otimes C$ is a functor, so it preserves the isomorphism.
But what if we consider the example, $\mathbb{Z}\otimes\mathbb{Z}_2$ is not isomorphic to $2\...
-2
votes
1answer
57 views
about flat $R$-modules with different conditions [closed]
Need to know if the following statements are true or false:
Let $S$ be a multiplicative set in $R$. Then $(S^{-1}R)[x,y,z]$ is always a flat $R$-module.
Let $I,J \subset R$ be proper ideals of $R$ ...
0
votes
1answer
21 views
is (A+B)/B=A/B true for A, B modules?
If I have $A$ and $B$ two modules, is the following reasoning true?
$(A+B)/B=\{a+b+B : a\in A, b \in B \}=\{a+B : a \in A \}=A/B$
I am just doubting it since it is weirdly similar to the second ...
4
votes
3answers
72 views
How to show that $\text{Hom}_R(A\times B ,M)\cong \text{Hom}_R(A,M)\times \text{Hom}_R(B,M) $ when $A, B$, and $M$ are $R$-modules?
I am working on the problem below.
Let $A,B$ and $M$ be $R-$mudules. Show that
(1) $\text{Hom }_R(A\times B,M)\cong \text{Hom }_R(A,M)\times \text{Hom }_R(B,M)$.
For $(1)$, I built a ...
2
votes
2answers
24 views
What is an appropriate generalization of this module/ring result
Consider $R=\mathbb{Z}$ as a $\mathbb{Z}$-module, i.e. an abelian group.
Then, every (nonzero) submodule of $\mathbb{Z}$ is isomorphic to $\mathbb{Z}$ (as $\mathbb{Z}$-modules).
I am curious what is ...
