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Questions tagged [modules]

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

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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 $X\cap Y=0$?
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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?
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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 ...
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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 (...
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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 $\...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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 $\...
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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)}=\...
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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 ...
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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 ...
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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},...
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1answer
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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}{\...
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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 ...
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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 ...
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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....
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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 ...
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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 $0 \to A \to B \to C \to 0$ ...
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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 \...
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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}{\...
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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 ...
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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'...
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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 ...
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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?
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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 ...
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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 \...
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1answer
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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 ...
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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 $F$ ...
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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 ...
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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 $\mathbb{Z}_{d}$
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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."
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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 ...
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1answer
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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 \...
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2answers
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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^...
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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 $n$ ...
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1answer
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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 ...
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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)\...
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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 $...
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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 ...
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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 ...
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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
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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-...
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2answers
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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\...
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1answer
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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 $R$ ...
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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 ...
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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 ...
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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 ...