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

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Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & ...
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1answer
38 views

Localization of a coherent module is coherent

I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent $A$-module $M$, its localization $M_{f}$ at $f\in A$ is a ...
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1answer
22 views

Can you always construct a homomorphism from a module into its dual?

Suppose we have some $R$-module $M$ with $R$-dual $M^\ast=Hom_R(M,\,R)$. Can we always construct an $R$-module homomorphism $M\to M^\ast$?
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41 views

Example of two modules M, N where the set of the $m\otimes n$ is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong ...
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34 views

Generalization of Singular locus and non-free locus to an algebra

Let $R$ be a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ and $ \Lambda $ be a noetherian $ R $-algebra. Recall that: (1) The singular locus of $R$, denoted by $\mathsf{Sing} ...
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1answer
28 views

A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\text{End}_k(V)$ be the ring of right linear transformations on $V$. My questions are: (1) Is $V$ a simple ...
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1answer
34 views

$Z_n$ as $Z[i]$-module

I am trying to find all $n$ such that $Z_n$ is a $Z[i]$-module, where $Z[i]$ is a Gaussian integer ring. I proved that any $Z[i]$-module $M$ is just an Abelian group with hommomrphism ...
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29 views

Under what conditions is a ZG-module torsion-free?

If we have a ZG-module A, I was wondering if there are known condition we may imply on either A or the group G to make A torsion-free?
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1answer
29 views

Difference between torsion and zero divisor

I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This ...
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1answer
72 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
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1answer
42 views

What's a diagonal sum of two matrices?

Let $A$ be an $n\times n$ matrix over a field $K$. Show that there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal sum of an invertible matrix and a nilpotent matrix. (Hint: use ...
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2answers
66 views

“If $x$ is a non-unit, then $1-ux$ is a unit”

I don't understand these two lines from my book. We are given that $R$ is a local ring. If every $2$-generator submodule is cyclic and $Ra$, $Rb$ are given, then $Ra+Rb=Rc$, hence ...
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1answer
33 views

What's a $2$-generator submodule?

Context: Let $R$ be a local ring and $M$ an $R$-module. Show that the set of all submodules of $M$ is totally ordered by inclusion iff every finitely generated submodule of $M$ is cyclic or, ...
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3answers
74 views

What does $\overline{r}m:=rm$ mean?

On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ What does that action actually mean? What is $\overline{r}$?
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22 views

Decompose finitely generated modules and use Krull-Schmidt theorem [duplicate]

I'm trying to show that if $R$ is an Artinian ring, then for finitely generated modules $M,N,N'$, we have that $M\oplus N\cong M\oplus N'$ implies that $N\cong N'$. I'm supposed to do this by ...
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2answers
112 views

Modules that have finitely many submodules

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector ...
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1answer
40 views

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
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0answers
27 views

Free and projective modules

I have two simple questions. Let $R$ be a ring. Is every free $R$-Mod of the form $R^{(S)}$ for some set $S$? Can $R^S$ be projective for an infinite set $S$, for suitable, non-trivial $R$? Note that ...
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1answer
97 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
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1answer
27 views

Simple bimodule over a basic algebra

I am looking for a reference for the following result: Let $k$ be an algebraically closed field and $A$ be a finite dimensional $k$-algebra. If $A$ is basic, then every simple $A$-$A$ bimodule is ...
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1answer
28 views

How can I show $I\simeq L$ where $L$ is a direct summand of $X$ such that $L\subseteq \textrm{ker}(g)$?

Consider the exact sequence of $R$-modules ($R$ is ring with identity $1_R$): $$ 0\longrightarrow K\oplus I\stackrel{f}{\longrightarrow} X\stackrel{g}{\longrightarrow} Y\longrightarrow 0,$$ where $I$ ...
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2answers
38 views

Equivalent definitions of faithful flatness

I am currently studying the equivalent definitions of faithful flatness over (probably noncommutative) unital rings. In particular, there is a version that I have doubts about: Let $R \subseteq S$ ...
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1answer
46 views

Three different notions for “bigness” of a module (length, rank and “mass”).

Let $M$ be an $R-$module where $R$ is an integral domain. I'm trying to understand the relations between these three notions of size: Let $S \subset M$ be a generating set of minimal cardinality. ...
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1answer
33 views

A question about 1.0.3 in Grothendieck's EGA

In (1.0.3), Grothendieck states that, given non-commutative rings $A$ and $B$, a homomorphism $\varphi : A \to B$, and a left ideal $\mathfrak{J}$ of $A$, the left ideal $B\mathfrak{J}$ of $B$ ...
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1answer
76 views

If $A$ is a finitely generated $R$-module, is $\operatorname{Hom}_R(A,R)$ finitely generated? [duplicate]

Let $R$ be an utterly arbitrary commutative, unital ring. Let $A$ be a finitely generated $R$-module. Is $\operatorname{Hom}_R(A,R)$ finitely generated as an $R$-module? Intuitively and based on ...
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0answers
34 views

Projective and injective resolutions of cyclic modules

Let $R$ be a ring and $M$ a cyclic $R$-module. It is well-known that always exist projective and injective resolutions of $M$. Is it any method to construct explicitly these resolutions which have the ...
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1answer
57 views

Module of constant rank over noetherian reduced ring

Let $A$ be a reduced noetherian commutative ring and $M$ be a finitely-generated $A$-module such that for all prime ideals $\mathfrak p$, $M_{\mathfrak p}/\mathfrak pM_{\mathfrak p}$ is an ...
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2answers
60 views

Why do nil ideals annihilate simple modules?

A nil ideal $N$ of a ring $R$ is defined as follows: $(N,+)$ is a subgroup of $(R,+)$ $\forall x \in N, \forall r \in R :\quad x \cdot r \in N$ $\forall x \in N, \forall r \in R : \quad r \cdot x ...
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1answer
22 views

Decomposition of Torsion Module

Let $k$ be a field, $k[X]$ the polynomial ring in one variable and $M$ a torsion $k[X]$-module (not necessarily finitely generated). Consider the submodules \begin{equation*} M_1 = \{a \in M \mid ...
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22 views

Looking for a Coordinate Free Way to Prove a Precursor to Nakayama Lemma.

Let $M$ be a finitely generated module over a ring (commutative with identity) $R$. Let $\mathfrak a$ be an ideal of $R$ and $\phi:M\to M$ be an $R$-module homomorphism such that ...
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37 views

Too Many Members in a Finitely Generated Module are Linearly Dependent

I am new to module theory and as of now am not very comfortable with the subject. So can somebody please check whether my claim and its proof is okay? Consider the following statement: Let $M$ be ...
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1answer
26 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
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3answers
30 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
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1answer
45 views

Tensor power modulo cyclic group action

Let $M$ be some $R$-module and $n \geq 1$ be some positive integer. The cyclic group $\mathbb{Z}/n\mathbb{Z}$, with a chosen generator $t$, acts on $M^{\otimes n}$ via $t(m_1 \otimes \dotsc \otimes ...
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1answer
29 views

If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right ...
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2answers
51 views

What is the significance of $A+ (B\cap C)=(A + B)\cap C$, where $A\subseteq C$, for modules?

My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here. Isn't ...
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0answers
55 views

Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
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0answers
53 views

Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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0answers
46 views

proving $\mathbb {Z}_n $ is an injective $\mathbb {Z}_n$-module [duplicate]

How can I prove a well-known fact: $\mathbb {Z}_n (=:\mathbb {Z}/n\mathbb {Z})$ is an injective $\mathbb {Z}_n $-module? I classified all ideals of $\mathbb {Z}_n $ and tried to use Baer's ...
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2answers
59 views

Flat Non Projective $A$-Module [duplicate]

A standard fact in Commutative Algebra is that a Projective $A$-module is flat. The converse is false. Can someone show me an example of a Flat Non Projective $A$-Module? Thank you!
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Yetter-Drinfeld modules over Hopf algebra

Let $H$ be a bialgebra and assume that $M$ is left $H$-module and left $H$-comodule. $M$ is called Yetter-Drinfeld module iff $\forall h\in H, \ \forall m \in M \ \ ...
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1answer
26 views

What is a composition series for $N$ and $M/N$

Let $M$ be a module with a composition series. Show that every submodule of $M$ is a member of a composition series for $M$. I understand the concept of a composition series for a module: it's a ...
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2answers
42 views

For an exact sequence $0\to M_1\overset{f_1}\to\cdots\overset{f_r}\to M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$?

For an exact sequence $0\to M_1\overset{f_1}\rightarrow\cdots\overset{f_r}\rightarrow M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$? $M_i$s are modules and $l(M_i)$ ...
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1answer
62 views

counterexamples on modules

Prove or disprove: for any comm. ring $R $, For three $R $-modules $M $, $N $ and $P $, $M \oplus P \simeq N \oplus P $ implies $M \simeq N $. Let $f:M\to N $ and $g:N\to M $ be two $R $-module ...
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1answer
58 views

$n+1$ elements in the free module $R^n$ must be linear dependent?

Suppose that $R$ is a commutative ring with unit and $R^n$ is the free module of dimension $n$ over $R$, it is right that $n+1$ elements in $R^n$ must be linear dependent?
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2answers
62 views

Example for an ideal which is not flat (and explicit witness for this fact)

I'm looking for an ideal $\mathfrak{a}$ of an commutative (possibly nice) ring $A$ together with an injective $A$-module homomorphism $M\hookrightarrow N$ such that the induced map ...
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1answer
38 views

Finite (cardinality) modules over a PID

Let $R$ be a principal ideal domain with $|R|=\infty$. Suppose $M$ is an $R$-module such that $2 \le |M| < \infty$. What properties does this imply about $R$? Background: I was hoping that $R\cong ...
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2answers
40 views

Are all injective endomorphisms of a module automorphisms?

As far as I understand, an automorphism is an isomorphism from a set to itself. If we have a homomorphism $f:M\rightarrow M$, then, from the first isomorphism theoreom, $im(f)$ is a submodule of $M$. ...
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1answer
43 views

Why are free modules called “free”?

Let $R$ be a ring (not necessarily commutative) with multiplicative identity. A $R$-module $M$ is called free if $M$ has a linearly independent generating set $\beta\subseteq M$. That is, for any ...
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1answer
291 views

Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...