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

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Modules over algebras vs Modules over Rings?

Since I've started module theory I was confused with a point. What is more general, the theory of modules over algebras or over arbitrary rings? I hope this is not a pointless question so let me try ...
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1answer
38 views

On selfinjectivity of Hopf algebras

Any group algebra $kG$ of a finite group is selfinjective. More generally Gentile proves that for a group ring $RG$ with $R$ commutative and torsion free as a $\Bbb Z$-module, $RG$ is selfinjective ...
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44 views

Find two generators of a $\Bbb{Q}[x]$-module $\Bbb{Q}^5$.

Let $T$ be a linear transformation on $\Bbb{Q}^5$ which is defined by $$T(v)= \begin{pmatrix} -1 & 0 & 0 & 0 & 3\\ 1 & 2 & 0 & -4 & 0\\ 3 & 1 & 2 & -4 &...
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33 views

Modules as morphisms to endomorphism rings

An $A$-module $M$ may be thought of as a (surjective) ring homomorphism $f: A \to E(M)$, where $E(M)$ is a ring of group endomorphisms of $M$. Then $am = f(a)(m)$. Is there any more to this ...
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1answer
58 views

Tensor fields on a manifold

Let $M$ be an $n$-dimensional smooth manifold. It is easily shown that the modules $\Gamma(TM)$ (the real vector space of vector fields on $M$) and $\Gamma(T^\ast M)$ (the real vector space of $1$-...
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19 views

An example of a module that have no supplement.

We see that if R/J is not coclosed coprojective and J has a supplement then R/J is projective. Now we are looking for an example J has no supplement and also R/J is not coclosed coprojective. But we ...
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106 views

Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
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89 views

Show that $R$ is a field

Let $R$ be a commutative ring with unit. If $R\neq 0$ such that each finitely generated $R$-module is free then $R$ is a field. In my notes there is the following proof: We need to show that ...
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28 views

How can we find a generating set?

How can we find a generating set of the $\mathbb{Z}$-module $\mathbb{Z}$ that does not contain the basis which is $\{1\}$ ? I saw in my notes that such a set is the $\{2,3\}$. Why is this a ...
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54 views

Show that $\text{Hom}_R(R^n,M)\cong \prod_{i=1}^n\text{Hom}_R(R,M)$

Let $R$ be a commutative ring with unit and let $M$ be a $R$-module. It holds that $\text{Hom}_R(R^n,M)\cong \prod_{i=1}^n\text{Hom}_R(R,M)$, right? How could we prove this? Do we have to define ...
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55 views

Finite type + integral = finite

Let $A \subseteq B$ be rings (comm. with unity). I am struggling to see why the following equivalence holds for $B$ interpreted as a $A$-Algebra: $A \rightarrow B$ is of finite type and $A\...
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33 views

groups as modules

Let $G$ a abelian group. Prove that $G$ is a $\mathbb{Z}$-module. Let $x\in G$ and $n\in\mathbb{Z}$ we define $xn$ as follows: If $n\geq 0$, then $x0=0$ and $x(n+1)=xn+x$. If $n<0$, then $xn=(-x)(...
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1answer
21 views

Empty singular submodule

I search for a module $M$ with its singular submodule $Z(M)$ the empty set, i.e. for every element $m$ of $M$ the annihilator of $m$ in $R$ is not essential, say, as right $R$-module; or proving that ...
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66 views

Extension of Scalars is well-defined

The reason I'm asking this, is because as an exercise, I'm asked to prove the following: Let $A$, $B$ be rings, $f:A\to B$ a ring homomorphism inducing $A$-module structure on $B$, and $M$ a flat $A$-...
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32 views

Prove that any finitely generated submodule of $R^+$ (the field of quotients) is free of rank $1$

I am working on the following problem: Let $R$ be a principal ideal domain and $R^+$ the field of quotients. Then $R^+$ is an $R$-module. Prove that any finitely generated submodule of $R^+$ is a ...
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1answer
22 views

Name for submodule killed by a right ideal

Let $\mathfrak a$ be a right ideal in a ring $R$. The set $N=\{m\in M: \mathfrak am=\mathfrak 0\}$ is a submodule of the left $R-$module $M$: If $m,n\in N$, $a\in \mathfrak a$, then $a(m-n) = am-an =...
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71 views

Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. $$V=\underbrace{\mathbb KG\oplus\cdots\oplus \...
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32 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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121 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
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17 views

Let $M$ be a finitely generated module over a P.I.D.. If $M=A\oplus B$, $A$ is a torsion submodule and $B$ is free, then $A=M_{\text{tor}}$.

I am reading the Goodman's Algebra. There is a lemma. Let $M$ be a finitely generated module over a principal ideal domain $R$. (a) If $M=A\oplus B$, $A$ is a torsion submodule, and $B$ is free, ...
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83 views

$\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N)$

Let $(M_i)_{i\in I}$ be a collection of $R$-moduls. Show that for all $N\in \text{Ob}(_R\text{Mod})$ is $$ \text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N). $$ My ...
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23 views

$R$-module strcture on $M\oplus A$ so that it becomes a $R$-algebra?

Let $R$ be a commutative ring with identity, $A$ an associative $R$-algebra without identity and $M$ a $A$-bimodule. I read I can endow $M\oplus A$ with a structure of $R$-algebra with the product $$(...
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56 views

primary decomposition of injective envelope of a module

The Exercise A3.6 of Eisenbud's book, Commutative Algebra with a view Toward Algebraic Geometry, is: Assuming that $R$ is Noetherian, let $M$ be any finitely generated $R$-module. a. Let ...
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32 views

Reducing a marix to Smith Normal form

I am trying to reduce the following matrix to Smith Normal form - $$ A= \begin{pmatrix} 1&0&0\\ 1&2&0\\ 1&0&3 \end{pmatrix}$$ What ever row and column operations I try, I end ...
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28 views

Singular ideal of an idealization

Let $S$ be a commutative ring, and let $A$ be a faithful $S$-module. Through idealization, we can make the abelian group $R=S⊕A$ into a commutative ring using the multiplication $(s,a)(s',a')=(ss',sa'+...
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34 views

Where am I going wrong in computing the cokernel of this map?

Let $A:\mathbb {Z^3\to Z^3}$ be the map given by $$A=\begin{pmatrix} 1&0&0\\ 1&2&0\\ 1&0&2 \end{pmatrix}$$ Then the cokernel of $A$ is $\mathbb Z^3/\text{ Im }A$ which can ...
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49 views

Inverse limit of flat algebras is flat?

Let $k$ be a field, let $R$ be a $k$-algebra, let $\{ S_i \}_{i \in I}$ be an inverse system of $k$-algebras, and let $R \to S_i$ be a $k$-algebra homomorphism making $S_i$ into a flat $R$-module. Is ...
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40 views

Proof of the exactness of the tensor product

In Atiyah and MacDonald, Prop 2.18 establishes that for any exact sequence $$M'\xrightarrow{f}M\xrightarrow{g}M''\xrightarrow{}0\tag{1}$$ of $A$-modules and homomorphisms, and for any $A$-module $N$, $...
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2answers
63 views

Proving/Disproving $M$ has the structure of an $R$-module

Given an abelian group $M$ and a ring $R$, how can one prove or disprove that $M$ has the structure of an $R$-module? When proving $M$ is an $R$-module, if it is not obvious how to define an action $R\...
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48 views

On boolean algebras as rings, modules and/or R-algebras

After trying to make sense of first order logic from an algebraic point of view I started to read about boolean algebras (similar to the explanations given here: wikipedia on boolean algebras. I also ...
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22 views

Element separated from a submodule by a homomorphism

Let $M$ be a module over a commutative ring $A$, $M'$ a submodule, and $y\in M\setminus M'$. Then $y$ can be separated from $M'$ by homomorphism. What I wish to prove is that there exists an $A$-...
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37 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
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28 views

About rank of free module compared to spanning and LI set

Let M be a free R-module where R is commutative ring with unity. Q1: If M is FG by n vectors. Will rankM $\leq$ n? Q2: If M has a LI set $S$. Will #$S \leq$ rankM? What if R is PID? Are above ...
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27 views

Writing a homomorphism in multiplicative form

Consider the additive group $\mathbb {Z_2}=\{0,1\}$ and consider the group $\mathbb Z_2^n$. Let $u_1,\ldots,u_d$, $(d>n)$ be not necessarily distinct , non-zero elements of $\mathbb Z_2^n$. Define ...
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18 views

Can we know what are the form of submodules of Z\oplus Z. [duplicate]

I know for instance Z(2,3), Z(2n,0) or Z(n,n) are submodules of the Z-module Z+Z. But can we know what are the submodules of Z\oplus Z exactly?
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15 views

Relation between the dual module and the dual of a vector space.

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite ...
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53 views

Why is a bilinear map $M \times N \to B$ the same as a homomorphism $M \to Hom_R(N,P)$?

Eisenbud's "Commutative Algebra" states that: It is elementary that a bilinear map $M \times N \to P$ is the same as a homomorphism $M \to Hom_R(N,P)$ While this makes some intuitive sense I keep ...
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27 views

A embedding of tensor product over semisimple algebras

Let $R$ be a semisimple Artinian algebra over the complex number field $\mathbb{C}$, that is, $R$ is isomorphic a finite direct product of matrix rings over $\mathbb{C}$. Let $S$ be a ideal of $R$, ...
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57 views

Compute the projective dimension of the given $R$-module

Let $$R=\frac{K[[x,y,z]]}{\left<xz,yz\right>}\text{ and } M=\frac{R}{\left<z+\left<xz,yz\right>\right>}.$$ Compute the projective dimension of $M$ as an $R$-module. My attempt ...
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38 views

Expressing an $R$-module as a direct sum

Let $R$ be a ring with $1$, $M$ a right $R$-module, and $f:M \to R$ a homomorphism of $R$-modules with $f(M)=R$. Then there is a decomposition $M=K \oplus L$ such that $f \vert_L:L\to R$ is an ...
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Finding the structure of an $F_p[X]$module

$p$ is a prime and $M$ is an $F_p[X]-$ module. Given $(X-1)^3M=0, \vert (X-1)^2M\vert=p, \vert (X-1)M\vert=p^3$ and $\vert M\vert =p^7$, determine $M$ as an $F_p[X]-$module, up to isomorphism. Using ...
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52 views

If $M_1$ and $M_2$ are distinct maximal ideals of, then $R/M_1 \otimes _R R/M_2=0$ [duplicate]

$R$ is a commutative ring with $1$ and $M_1,M_2$ are distinct maximal ideals of $R$. Then $R/M_1 \otimes _R R/M_2=0.$ Consider $f:R/M_1 \times R/M_2\to R/M_1 \otimes _R R/M_2 $ defined by $f(r+M_1,...
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26 views

Alternative proof : Group algebra contains all irreducible G-modules.

It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
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1answer
34 views

Equivalent definitions of semisimplicity of finite dimensional modules

Let $A$ be a finite dimensional $k$-algebra, and let $V$ be a finite dimensional $A$-module. How do we show that $V$ is semisimple (i.e. is the direct sum of simple submodules) if every maximal ...
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45 views

Modules over $\mathbb Z/p^n\mathbb Z$ have $p$-power order

I am given that $M$ is a module over $\mathbb Z/p^n\mathbb Z$, where $p$ is a prime and $M$ is finite. How can I show that the order of $M$ is $p^j$ for some $j$? The ring $Z/p^nZ$ is not a PID ...
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84 views

Let $\phi: M \to N$ be an $R$-module homomorphism having a left-inverse $\psi$. Prove that $\ker (\psi) \cong \operatorname{coker} (\phi)$ [closed]

Let $\phi: M \to N$ be an $R$-module homomorphism having a left-inverse $\psi$. I need to prove that $\ker ( \psi )\cong \operatorname{coker} \phi $. How to approach the problem? Any hints will be ...
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62 views

Understanding a certain proof about $R$-module homomorphisms and split exact sequences

I'm currently reading "Algebra: Chapter 0" by Paolo Aluffi. Before I state my problem, I would like to give here the definition of split exact sequences from the book: A short exact sequence ...
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45 views

Minimal spanning set $=$ basis

For finite dimensional vector spaces over a field it is true that if I have a spanning set whose cardinality equals the dimension of the vector space then that spanning set is a basis. Is this theorem ...
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1answer
24 views

Let $M$ be a nonzero cyclic module. $M$ is simple iff $\mathrm{Ann}(M)$ is a maximal left ideal

I'm struggling a bit with the proof of this statement. What I have is that, letting $M$ be a module over the ring $R$ (not necessarily commutative), $\exists \,x\in M$ s.t. $M=Rx$. Now, using the ...
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1answer
85 views

$A^\times/k^\times$ is a free $\mathbb{Z}$-module of rank of at most $r - 1$

Consider an algebraically closed field $k$, a finite extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, the integral closure $A'$ of $k[1/T]$ in $K$, and the integral closure $A''$ of ...