# Questions tagged [modules]

For questions on modules over rings.

459
questions

**4**

votes

**1**answer

105 views

### Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor:
$$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$
left-adjoint to the (forgetful) embedding:
$$...

**1**

vote

**1**answer

160 views

### Indecomposable weirdos (cnt.)

This post is a continuation of Weirdos but algebraic.
Logically, the quoted post could follow the present one rather than precede it.
Question Does there exist an indecomposable weirdo which is ...

**1**

vote

**0**answers

54 views

### On submodules of vector fields

I don't know much about modules aside from their basic definition and that they are more complicated than vector spaces.
I am asking this question because I wish to have a more "algebraic" ...

**0**

votes

**0**answers

69 views

### Certain morphism between graded modules

Say we have a morphism $f=(f_{i,j}) : \oplus_{i}M_{i} \rightarrow \oplus_{j}N_{j}$ (The direct sum of graded modules is finite) and $f_{i,j}$ are morphisms of graded modules but not necessary with the ...

**1**

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**0**answers

44 views

### Existence of nontrivial transfinite divisibility in $R$-modules

Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...

**0**

votes

**1**answer

51 views

### Can we extract an injective envelope from a monomorphism?

Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we ...

**3**

votes

**1**answer

67 views

### Effect of extending scalars on maps of modules

Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \...

**1**

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**1**answer

93 views

### Geometric meaning of colocalization of modules?

Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...

**4**

votes

**1**answer

103 views

### Global splitting field for algebras

Let $A$ be a finite dimensional algebra.
A field $K$ is a splitting field for an indecomposable $A$-module $M$ in case the local algebra $End_A(M)/(rad(End_A(M))$ is 1-dimensional.
$K$ is called a ...

**4**

votes

**1**answer

124 views

### Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...

**2**

votes

**0**answers

209 views

### Module structure for $\mathbb{Z}$

I am interested to know which module structures we can define in the additive group of integers $\mathbb{Z}$.
It is easy to prove that $\mathbb{Z}$ does not admit a vector space structure. (For more ...

**4**

votes

**1**answer

179 views

### Linear algebra over non-commutative semirings

I'm reading up on linear algebra over semirings, and I'm wondering why people seem to stop short of showing an equivalence between linear transformations between free modules and matrices.
It seems ...

**5**

votes

**1**answer

124 views

### Projective module which splits off sequence of submodules, but not the sum

Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that:
$X$ is projective,
$X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...

**3**

votes

**0**answers

123 views

### The category $\text{add}M$ for a module $M$

Let $A$ be a $K$-algebra and consider the category $\text{mod}A$ of finitely generated left $A$-modules. If $M$ is in $\text{mod}A$, then $\text{add}M$ denotes the full subcategory of $\text{mod}A$ ...

**2**

votes

**1**answer

165 views

### Left module which cannot be made into a bimodule?

Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...

**2**

votes

**0**answers

44 views

### Non-singular rings which are Rickart

A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$.
It turns out that a ring $R$ is right Rickart iff every ...

**0**

votes

**1**answer

279 views

### A lemma on a sequence of three morphisms

Few months ago, I proved that when there is three morphisms of modules over a commutative ring with zero composition, i.e., a sequence
$$A \xrightarrow{\alpha} B \xrightarrow{\beta} C \xrightarrow{\...

**1**

vote

**1**answer

128 views

### Injective hull of quotient of modules

Let $R$ be a Noetherian commutative ring. If $N\leq M$ are $R$-modules, then is $E(M)$ isomorphic to a submodule of $E(N)\oplus E(M/N)$?
Here $E(M)$ denotes the injective hull of $M$.

**4**

votes

**2**answers

263 views

### Adjoints for radical and socle functors

Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...

**0**

votes

**1**answer

130 views

### Is a coslice (slice under) category a full subcategory of it ambient category?

Let $\operatorname{C}$ be a category and $c\in \operatorname{C}$ an object. Consider the coslice (sometimes called slice under) category ${\operatorname c}/{\operatorname C}$. My question is whether ${...

**9**

votes

**1**answer

188 views

### Is an overring of an order reflexive as a module over the order?

Let $R$ be a one-dimensional Noetherian domain with fraction field $K$, let $\tilde{R}$ be the integral closure of $R$ in $K$, and assume that $\tilde{R}$ is finitely generated as an $R$-module. (In ...

**3**

votes

**0**answers

135 views

### Is there a constructive proof of Baer's Criterion?

Baer's Criterion states than one can check injectivity of an $R$-module on inclusions of ideals. The proof, however, strikes me as very nonconstructive: it employs both Zorn's Lemma and LEM.
Does ...

**0**

votes

**1**answer

120 views

### Notion of module over commutative post lie algebra

Let $ (S, \{. \}, [.]) $ be an algebra over a vector
space endowed with two bilinear maps $ \{. \}, [.] : S \times S \rightarrow S$ and satisfying some compatibility conditions.
In example, if S is ...

**-4**

votes

**1**answer

97 views

### Finitely generated module [closed]

If a finitely generated module $M$ injected in a free module, then would the image of $M$ be a free finitely generated module?

**7**

votes

**2**answers

559 views

### Beauville-Laszlo for schemes

For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...

**2**

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**0**answers

65 views

### Lattices with trivial coinvariants for finite groups

Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank.
Question: Is there a finite group $G$ and a $\mathbb{Z}...

**6**

votes

**3**answers

516 views

### Dual of a bimodule

For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module.
Note: ...

**4**

votes

**1**answer

461 views

### Parabolic Kazhdan-Lusztig Conjecture

In this post, Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^*$, which is under the name: Kashiwara and Tanisaki - Characters of ...

**9**

votes

**0**answers

173 views

### What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...

**7**

votes

**1**answer

246 views

### Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...

**4**

votes

**0**answers

139 views

### Row rank and column rank of matrix with entries in a commutative ring

Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed
in "Ranks of Modules"
one can say that the row rank of $A$ is ...

**0**

votes

**0**answers

175 views

### Partial monoid in the category of categories of modules: The spotty nature of monad composition

It seems that I am working on a conjecture in category theory. In particular, I am curious about the spotty nature of the composition of monads on Set. I am guessing that there is a category, $\...

**2**

votes

**0**answers

80 views

### Isoartinian and isosimple modules

I'm reading this article by A. Facchini and Z. Nazemian, in wich they discuss modules with chain conditions up to isomorphism. A couple of the main concepts are the following:
Definition
We say that ...

**1**

vote

**1**answer

102 views

### Hom functor and arbitrary coproducts

Let $M$ be a finitely generated $R$-module. It's easy to check that, in this case, $\mathbf{Hom}(M,-)$ preserves infinite sums. Now suppose that $M$ is projective. Is the
reciprocal true? That is, if ...

**2**

votes

**0**answers

90 views

### Fock space for distinguishable particles

I am proposing that a category, $\mathcal{C_{\mathcal{Fd}}}$, of Fock Spaces for distinguishable particles is, in fact, a suitably defined tensor product in the category of categories of Modules. In ...

**-2**

votes

**1**answer

138 views

### Module such that every finitely generated submodule is semisimple [closed]

Is there an example of a module $M$ (over a commutative ring) that is not free, and such that each of its finitely generated submodule is semisimple (i.e. such that any submodule of any finitely ...

**0**

votes

**1**answer

35 views

### Equivalent condition for distributive property of $\mathcal{S}(M)$ [closed]

$\textbf{Definitions:}$
Let $M$ be an $A$-module. The collection $\mathcal{S}(M)$ of all submodules of $M$ is partially ordered by the inclusion relation. The collection $\mathcal{S}(M)$ is said to ...

**1**

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**0**answers

92 views

### Composition of monads induces tensor product in the category of modules

I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the ...

**5**

votes

**2**answers

466 views

### Zero tensor product over a complex algebra?

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.
Q1: Can this ...

**3**

votes

**1**answer

297 views

### When does the forgetful functor from modules to vector spaces have a right adjoint?

Given any algebra $R,$ when does the forgetful functor
$R\text{-}Mod \rightarrow Vec$
have a right adjoint?
Does this imply any finiteness conditions on R?
Is there a book/paper discussing this?
I'...

**0**

votes

**0**answers

80 views

### Structures like vector spaces but closed under heterogeneous products

The category of (pseudo-)Euclidean vector spaces (vector spaces with a nondegenerate but not necessarily positive-definite quadratic form) is not closed under products because $R^n$ over $R$ and $Z_2^...

**2**

votes

**0**answers

89 views

### What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...

**2**

votes

**1**answer

228 views

### Flatness of submodules of free modules

Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group.
If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...

**7**

votes

**2**answers

443 views

### Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$.
Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...

**7**

votes

**1**answer

322 views

### Global to local principle for f.g. $\mathbb{Z}[x]$ modules

In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...

**4**

votes

**0**answers

125 views

### When does the canonical $t$-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...

**2**

votes

**1**answer

143 views

### Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...

**2**

votes

**1**answer

110 views

### A weak Schur's lemma for non-semisimple finite dimensional algebras

Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be
a decomposition of $B$ into ...

**5**

votes

**0**answers

105 views

### Any f.p. faithful simple module over a primitive group ring?

Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...

**3**

votes

**0**answers

116 views

### What is the $Ass(Ext^p_R(M,R))$?

Let $R$ be a Noetherian commutative local ring, $M$ a finitely generated $R$-module with $p=pd M<\infty$ (projective dimension of $M$). What is the relation between $Ass(Ext^p_R(M,R))$ and $Ass(M)$?...