All Questions
Tagged with modules ra.rings-and-algebras
195 questions
3
votes
0
answers
292
views
modules over principal ideal rings
Let $R$ be a commutative principal ideal ring (not necessarily Artinian) and let $M$ be a finitely generated $R$-module. Is $M$ a direct sum of cyclic $R$-modules? (i.e. a generalization of the theory ...
- 429
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 12.9k
5
votes
0
answers
93
views
Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules
Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
- 429
2
votes
1
answer
393
views
When the annihilator of each nonzero submodule is prime
Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...
- 21
3
votes
1
answer
270
views
Thin representations for quiver algebras
A representation $M$ of a quiver is called thin when $M$ has a dimension vector consisting only of 0 or 1 entries.
When $A=kQ$ is a path algebra for a tree $Q$, then there is the nice result that ...
- 356
3
votes
1
answer
299
views
RIng that is flat over a subring as a right module but not as a left module
What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module.
The following question is my motivation:
Faithful flatness for ...
- 10.8k
4
votes
0
answers
363
views
A projective module over a domain that is not faithfully flat?
Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
- 1,766
2
votes
2
answers
181
views
Module complements to rings embedded in a projective module
Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module.
A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-...
- 38.6k
1
vote
0
answers
134
views
Composition of faithfully flat ring extensions
Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
- 435
3
votes
2
answers
287
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" ...
- 26.3k
1
vote
0
answers
99
views
Finding an injective envelope containing another injective envelope
Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
- 12.9k
3
votes
1
answer
277
views
Faithful flatness for rings
Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
- 38.6k
14
votes
1
answer
581
views
Group rings such that every (countably generated) module has a maximal submodule
Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
...
CommunityBot
- 1
2
votes
1
answer
287
views
Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable
Crossposted on StackExchange on July 28 (no answer so far).
Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
- 18.7k
2
votes
2
answers
256
views
An algebra which is a direct sum of simple sub-bimodules over a subalgebra
Let $A$ be an infinite-dimensional noncommutative algebra over a field, let $B$ be an infinite-dimensional subalgebra of $A$, and let $A$ be a direct sum of projective simple $B$-sub-bimodules. Then ...
- 38.6k
2
votes
0
answers
252
views
Jacobson density theorem and Riesz representation theorem
My question is propably too vague to answer, the reader is advised.
There are these two mentioned theorems which you may recognise in other forms, but these are the ones that I have been taught.
...
- 1,848
4
votes
1
answer
669
views
When every proper submodule of a free module is contained in a maximal submodule
Let $(R, m)$ be a commutative local ring (it is not Noetherian in general) and $F$ be a free $R$-module. Under what conditions every proper submodule of $F$ is contained in a maximal submodule.
- 38.6k
6
votes
1
answer
541
views
Functors between module categories that comes from restriction
Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod $ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space.
...
- 38.6k
0
votes
0
answers
150
views
How to classify rings by combinatorial structures?
There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
1
vote
0
answers
73
views
Faithfull flatness of a module containing the ring as a direct summand
Let $R$ be a not necessarily commutative ring, and let $M$ be a projective left $R$-module.
Question. If $R$ is a direct summand of $M$ as a left $R$-module, then is it
true that $M$ is faithfully ...
- 393
2
votes
1
answer
197
views
Top and bottom composition factors of $M$ are isomorphic
Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...
CommunityBot
- 1
7
votes
1
answer
966
views
When is the category of finitely presented modules abelian?
Let $R$ be an associative ring with identity and $\mathrm{mod}R$ be the category of finitely presented $R$-modules. I would like to know when the category $\mathrm{mod}R$ is abelian.
I know that if $R$...
- 28.7k
0
votes
0
answers
110
views
Decomposition an $A$-module to irreducible ones
Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra.
Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
- 4,058
32
votes
7
answers
4k
views
"Sums-compact" objects = f.g. objects in categories of modules?
Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...
- 53.3k
6
votes
1
answer
190
views
Finitely presented modules admitting projective covers
A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ...
- 35.2k
5
votes
1
answer
132
views
Integral monoid rings and Ore conditions
Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.
I have two, ...
- 18.7k
2
votes
1
answer
160
views
MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
- 37.4k
3
votes
1
answer
186
views
If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?
EDIT: The question was originally about general Noetherian rings instead of PID's. Thanks to YCor for pointing out how wrong this was in the comments below (1 2 3).
Question 1: Let $R$ be a PID. ...
- 12.9k
2
votes
1
answer
290
views
Endomorphism rings of infinitely generated free modules generated by idempotents?
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
- 63.9k
1
vote
0
answers
21
views
Weakening s-unitality
Recall that a (non-unital, non-commutative) ring $R$ is left s-unital if for every $r\in R$, we have $r\in Rr$.
Consider the following conditions:
There is a nonzero integer $m$ such that for all $r\...
- 1,338
6
votes
0
answers
235
views
How much does Ext tell me about isomorphisms?
So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
- 14.1k
1
vote
1
answer
123
views
References about transfinite socle series
I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series.
Let $R$ be an associative unital ring and $...
- 35.2k
2
votes
0
answers
50
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 ...
- 63.9k
-1
votes
1
answer
67
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 ...
- 1,690
4
votes
1
answer
135
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 ...
- 35.2k
4
votes
1
answer
159
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 \...
- 26.5k
6
votes
1
answer
453
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 ...
CommunityBot
- 1
2
votes
0
answers
359
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 ...
- 63.9k
6
votes
2
answers
2k
views
Direct sum of injective modules is injective
By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an ...
- 63.9k
8
votes
1
answer
1k
views
Direct sum of injective modules over non-Noetherian rings
By the Bass-Papp theorem, if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists an injective module over $R$ non-Noetherian, that ...
- 63.9k
4
votes
1
answer
382
views
Non-existence of projective covers
I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at:
http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf
In ...
- 1,726
5
votes
1
answer
173
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 \...
- 858
3
votes
0
answers
268
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 $\text{mod}A$ ...
- 63.9k
3
votes
1
answer
244
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 ...
- 12.3k
2
votes
0
answers
56
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 ...
- 1,065
0
votes
1
answer
147
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 ...
- 181
10
votes
3
answers
1k
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: ...
- 7,746
7
votes
0
answers
657
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 ...
- 63.9k
3
votes
2
answers
227
views
Operations on semi-hereditary rings
I am learning about left (right) semi-hereditary rings, and got stuck with the following questions. Just to recall that being left semi-hereditary means that every finitely generated submodule of a ...
- 1
3
votes
2
answers
831
views
Divisible torsion $\mathbb{Z}$-modules
I am trying to prove that for any divisible torsion $\mathbb{Z}$-module $V$,
this map
$$f:\mathbb{Q}/\mathbb{Z}\otimes_E\text{Hom}(\mathbb{Q}/\mathbb{Z},V)\longrightarrow V\mbox{ defined by }
f((q+\...
- 63.9k