# Questions tagged [modules]

For questions on modules over rings.

652
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### If $R$ is a regular square-free ring, then $R$ is dual-square-free

Let $R$ be a ring with unity. We call $R$ regular if every principal (left) right ideal is a summand (i.e., generated by an idempotent). A module $M$ is called square-free if whenever $A,B$ are ...

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### A question on clean rings

Recall that a ring $R$ is called clean if every element of $R$ is a sum of a unit of $R$ and an idempotent of $R$. We call a module $M$ clean if its endomorphism ring $End(M)$ is a clean ring. ...

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### Lemma of Harada and Sai on sums of modules with a "chain" of monomorphisms between them

I am trying to get a contradiction from the following set of hypotheses:
Let $R$ be a ring. Let $M$ be a direct sum of non-zero $R$-modules $M_1$, $M_2$, $\dotsc$. For each $i\ge1$, let $f_i:M_i\to M_{...

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### What is the natural module?

Lemma 2.9 of [1]:
Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...

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### How fast does the number of "fixed" points grow compared to the size of the ball in the following group?

I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...

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### Characterize rings $R$, such that the countable product $P=R^N$ has the property that every finitely generated submodule of $P$ is free

What are the rings whose countable power has the property that every finitely generated submodule is free?

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### A commutative ring with unity which does not have relatively pseudo-injective ideals with zero intersection

Let $R$ be a ring with $1$ and $M$ and $N$ be any right $R$-modules. We say that $M$ is pseudo-$N$-injective if every $R$-monomorphism $f:X \to M$ from a submodule $X_R$ of $N_R$ can be extended to $N$...

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### Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$

Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...

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### Bounding the length in a module of evaluated skew polynomials

Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...

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### How to find the annihilator of a module/how to determine the torsion module

Given a left $R$-module $M$. Given a subset of $S\subset M$.
What are related theorems concerning the annihilator of $S\subset M$ that can be used in practice? $S$ is known.
In my case, $R$ is ...

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### When is a submodule trivial?

I am a beginner concerning module theory, but I need it for my PhD. Sorry for obvious questions therefore.
Given a left $C(G)$-module $(V, \tilde{\rho})$ where $C(G)$ denotes the group algebra over a ...

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### submodules in a direct sum of semisimple modules without common simple factors

Let $A$ be an associative (unital) algebra. Let $M_1,\cdots, M_r$ be pairwise non-isomorphic simple $A$-modules and let $V=\bigoplus^r_{i=1}V_i$, where
$$
V_i=M_{i,1}\oplus \cdots\oplus M_{i,n_i}\...

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### What is known about finite dimensional modules over the nilCoxeter algebra?

Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...

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### Endomorphism of torsion points of Drinfeld modules

Reposting from mathstackexchange.
A Drinfeld module is defined to be an $\mathbb F_q$-algebra morphism $\phi: \mathbb F_q[T] \rightarrow K\{\tau\}$, where $K=\mathbb F_{q^m}$ is a finite field and $K\{...

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### Factoring through projective modules is an equivalence relation

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PHom{PHom}$I'm reading about stable module categories, and I have a question about the definition of the maps. Let $R$ be a ring, and take (left) ...

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### Monoidal structure on presheaves

I am confused about the following monoidal structure, which gives a symmetric monoidal structure on R-modules (that I think is not Cartesian), even if R is not commutative.
Let $C$ be a small category....

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### Auslander-Reiten sequences where irreducible morphisms are all epi/mono

Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles.
Let $M$ be such a ...

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### Is the matrix ring $\mathbb{M}_n(R)$, $n\geq 2$, over a serial ring $R$ again serial?

Let $R$ be a ring with $1$. A right $R$-module $M$ is called uniserial if its submodules form a chain, i.e., for any two submodules $A,B\subseteq M$ either $A\subseteq B$ or $B\subseteq A$. The module ...

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### Infinite radical ideal cubed equals zero for tame hereditary Artin algebras

Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...

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### Generating sets for a module and scalar extension

Let $k$ be an algebraically closed field and $K/k$ a (transcendental) field extension. Let $A$ be a finite dimensional $k$-algebra, and $M$ an $A$-module. Suppose that the $K \otimes_k A$-module $K \...

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### Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?

All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...

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### $R$-Module objects in cartesian closed categories

I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...

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### Algorithms to decompose a graded module over $R[x]$, where $R$ is a PID

There is a certain class of objects, which can be thought of either as modules over a ring $R[x]$ or as functors. A few equivalent definitions are given below. The question is what computer algorithms ...

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### Length of dual module

It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...

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### Tensor product of modules in model vs. infinity categories

Let $C$ be a combinatorial symmetric monoidal model category and let $A$ be a associative algebra object in $C$, that is cofibrant as an object in $C$. In Higher Algebra 4.3.3.17, Lurie proves an ...

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### Totally isotropic space for bilinear pairing over ring

A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...

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### Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$

Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...

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### Is the span of all nilpotent ideals also a nilpotent ideal?

Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...

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### A question about freeness of a certain class of abelian groups

Lets call an abelian group $G$, to be semi-free (or SF) if every nonzero subgroup of $G$ is isomorphic to $\mathbb{Z}\times H$ for some abelian group $H$.
Is every semi-free group, a free group? If ...

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### Two nonsingular non-isomorphic modules with isomorphic injective hulls

Let $R$ be a ring with unity. Are there two nonsingular non-isomorphic right $R$-modules with isomorphic injective hulls?!.

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### Simple-direct-injective modules

A right $R$-module $M$ is called a simple-direct-injective module if it satisfies any of the following equivalent conditions:
For any simple submodules $A,B$ of $M$ with $A \cong B \subseteq^{\oplus} ...

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### When do limits of $R$-modules commute with direct sum?

Let $R$ be a commutative ring. Is there any good special case in which I can say that a limit of $R$-modules commutes with direct sum? This is of course true for finite direct sums. Are there other ...

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### An example of module which is square-free, CS, NOT C3, and NOT nonsingular

Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every ...

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### Trying to decode a module functor

This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten.
Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...

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### Integral closure in the total ring of fractions of reduced ring. Is it finitely generated?

Let $R = \bigoplus_{i = 0}^\infty R_i$ be a reduced finitely generated graded $\mathbb{C}$-algebra, $R_0 = \mathbb{C}$. Let $\overline{R}$ be the integral closure of $R$ in its total ring of fractions....

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### The eventual number of generators of modules of which $M$ is a subquotient

Let $R$ be a (commutative) ring and let $M$ be an $R$-module. Say that $M$ is subfinitely generated if $M$ is a submodule of a finitely-generated module. Write
$$\mathcal F(M) = \{ M \rightarrowtail N ...

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### Do graded-commutative rings satisfy the strong rank condition?

Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$.
It is ...

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### Connections in non-commutative geometry

Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...

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### The change-of-monoid adjunction between categories of modules induced by a morphism of monoids

Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...

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### Schur functors = Weyl functors in characteristic zero?

I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....

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### When is the intersection of a family of CS-modules is again CS?

Recall that a right module $M$ over a ring $R$ (with unity) is called CS if every submodule of $M$ is essential in a summand of $M$.
Let $\lbrace M_i \rbrace_{i\in I}$ be a family of right CS-modules ...

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### When is a module a filtered colimit of finitely presented submodules?

For a (commutative, say) ring $R$, and an $R$-module $M$ it is known that $M$ is both:
a filtered colimit of finitely generated $R$-submodules (by considering all finite subsets of $M$ and ...

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### Two exact sequences for $R$-modules: does one imply the other?

Consider the following two properties for an $R$-module $M$:
For every endomorphism $f:M\rightarrow M$, there exist central endomorphisms $g, h\in \operatorname{End}_R(M)$ such that the sequence $M_{...

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### Estimation for dimension of support and associated primes of a module of depth zero

Let $(R,\mathfrak{m})$ be a local Noetherian ring and $M$ a finitely generated $R$-module of depth zero, ie $\operatorname{depth}(M):=\text{depth}_{\mathfrak{m}}(M)=0$.
Can we make some interesting ...

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### For a pure-injective module $M$ does the property "$\operatorname{Hom}(-,M)$ is surjective" commute with certain limits?

$\DeclareMathOperator\Hom{Hom}$Let $M$ be a pure-injective module. Then $\Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\...

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### A general theory of pairings

Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra.
There are also text books for bilinear forms and related quadratic ...

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### Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)

Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...

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### Trivial group cohomology induces trivial cohomology of subgroups

From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...

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### Nomenclature help: action vs. module and “pointed” monadic algebras are module objects?

I'm confused about some nomenclature in a paper by J. Goguen from 1975. I'm happy to use definitions however they appear. But I aim to summarise the paper for a non-expert audience and I want them to ...

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### Shouldn't $\mathrm{End}_{C(TM)}(E)$ be defined differently in Heat Kernels and Dirac Operators?

The first four chapters of the book lead up to the proof of theorem 4.1. Its main consequence is that it provides the local index theorem for Dirac operators. The statement of theorem 4.1 involves a ...