Questions tagged [modules]

For questions on modules over rings.

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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 ...
Marty's user avatar
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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 ...
Hussein Eid's user avatar
2 votes
2 answers
123 views

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 ...
kevkev1695's user avatar
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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 \...
bm3253's user avatar
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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 ...
kevkev1695's user avatar
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3 votes
2 answers
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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 ...
GHPR's user avatar
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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 \...
JBuck's user avatar
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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 ...
Jakob's user avatar
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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 ...
JBuck's user avatar
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1 answer
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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 ...
Hebe's user avatar
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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 $k\in\mathbb{N}$ ...
Sanae Kochiya's user avatar
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1 answer
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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 ...
Mostafa's user avatar
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2 votes
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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?!.
Hussein Eid's user avatar
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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} ...
Hussein Eid's user avatar
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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 ...
Vik78's user avatar
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1 answer
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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 ...
Hussein Eid's user avatar
2 votes
0 answers
141 views

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 ...
Ilk's user avatar
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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....
user197284's user avatar
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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 ...
Tim Campion's user avatar
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8 votes
1 answer
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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 ...
Tim Campion's user avatar
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276 views

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. ...
Qwert Otto's user avatar
7 votes
1 answer
142 views

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 ...
Angelos's user avatar
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5 votes
1 answer
210 views

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....
Sunny Sood's user avatar
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39 views

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 ...
Hussein Eid's user avatar
7 votes
1 answer
306 views

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 ...
Jakob's user avatar
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1 vote
1 answer
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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_{...
Najmeh Dehghani's user avatar
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0 answers
60 views

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 ...
user267839's user avatar
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1 vote
1 answer
120 views

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 $\...
kevkev1695's user avatar
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1 vote
0 answers
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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 ...
Thomas Preu's user avatar
5 votes
0 answers
130 views

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 ...
Gro-Tsen's user avatar
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8 votes
1 answer
440 views

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 ...
Andrea Antinucci's user avatar
2 votes
1 answer
66 views

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 ...
Timtro's user avatar
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0 answers
132 views

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 ...
Filippo's user avatar
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2 votes
1 answer
102 views

Can we do away with cotensors when exploring the equivalence between closed $\mathscr{V}$-modules and strongly tensored $\mathscr{V}$-categories?

$\newcommand{\M}{\mathcal{M}}\newcommand{\ML}{\underline{\mathcal{M}}}\newcommand{\N}{\mathcal{N}}\newcommand{\NL}{\underline{\mathcal{N}}}\newcommand{\V}{\mathscr{V}}\newcommand{\VL}{\underline{\...
FShrike's user avatar
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2 votes
0 answers
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Pairs of ideals in an abelian category similar to torsion pairs

Let $\mathcal{A}$ be an abelian category. In the context of my work I am considering pairs of ideals $(\mathcal{I}, \mathcal{J})$ in $\mathcal{A}$ with the following properties: $\quad \mathcal{I} \...
kevkev1695's user avatar
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11 votes
1 answer
152 views

Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?

A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that $$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$ An example of such an object is ...
shin chan's user avatar
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1 vote
1 answer
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A characterization of semiartinian modules

Recall that a right module $M_R$ is called semiartinian if every nonzero homomorphic image has nonzero socle. It's well known that the following two statements are equivalent: $M$ is semiartinian. ...
Hussein Eid's user avatar
2 votes
0 answers
71 views

Are the following two characterisations of symplectic modules, using the language of form rings, the same?

Page 205 of the book Classical Groups and Algebraic K-Theory defines a symplectic module to be an arbitrary quadratic module $(M,h,q)$ over a form ring $(R,\Lambda)$ with $(J,\varepsilon)$ where $J=\...
wlad's user avatar
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1 vote
0 answers
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Is there any point in considering Form Rings when 2 admits an inverse?

In the study of quadratic spaces over general rings, there is a type of scalar which people consider called a Form ring $(R,\Lambda)$ relative to some anti-automorphism denoted $(-)^J:R\to R$ and ...
wlad's user avatar
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3 votes
2 answers
331 views

Zeros of higher Ext functors

I'm trying to understand module classes that are defined as the kernels of higher Ext functors (e.g., arising here; as this paper suggests, I'm coming at this problem outside of module theory). One ...
Will Boney's user avatar
3 votes
1 answer
306 views

A generalisation of induced representations

Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define: $W^G=\sum_{...
semisimpleton's user avatar
2 votes
0 answers
75 views

Almost split sequences for symmetric algebras

Let $k$ be an algebraically closed field and $A$ be a symmetric algebra. I want to know how to compute almost split sequences ending at a non-projective indecomposable right $A$-module $X$. Question: ...
sola's user avatar
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2 votes
1 answer
95 views

Baur-Monk quantifier elimination (BG-invariants in 1-free variable)

$\DeclareMathOperator\Inv{Inv}$Baur-Monk quantifier elimination implies that a sentence in the language of modules is a combination of BG invariant statements. A BG invariant sentence is a boolean ...
Eladio Vuente's user avatar
2 votes
1 answer
120 views

Question on Baur-Monk quantifier elimination for modules

Baur-Monk quantifier elimination theorem asserts that any formula in the language of modules is modulo the theory a boolean combination of BG-Invariants and positive primitive formulas. However, in p....
Eladio Vuente's user avatar
2 votes
1 answer
97 views

What are the conditions that can be given to a right $R$-module $M$ which makes $E(M)$, its injective hull, Artinian

Let $R$ be a ring with unity. Let $M$ be a unitary right $R$-module. It's known that a simple right module over a commutative Noetherian ring has an Artinian injective hull. I wonder what are the ...
Hussein Eid's user avatar
4 votes
1 answer
169 views

Exact sequences with two FL-modules

Let $R$ be a ring. An $R$-module $M$ is called FL (FP) if it has a finite resolution consisiting of finitely generated free (projective) modules. Given an exact sequence of $R$-modules, $0\to M_1\to ...
Andrei Jaikin's user avatar
1 vote
0 answers
44 views

Intersection of (sub-)modules under Laurent and formal rings

I have a (hopefully quick) question regarding an intersection of two tensor modules. Let K be a field and $A,B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
mathieu_matheux's user avatar
3 votes
0 answers
60 views

Automorphisms of matrix algebras and Picard group

This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE). Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
GreginGre's user avatar
  • 1,649
3 votes
1 answer
99 views

Does $M$ satisfy the descending chain conditions on $\mathbb{Z}G$-retracts?

‎Let $H$ be a subgroup of $G$‎. ‎Then a homomorphism $r:G\to H$ is said to be a retraction if the inclusion homomorphism $i:H\hookrightarrow G$ is a right inverse of $r$‎, ‎i.e‎. ‎$r(x)=x$ for all ...
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