# Questions tagged [modules]

For questions on modules over rings.

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### What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?

Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...
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### Modules with special properties

Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $End_A(M)$ annihilates the socle $soc_A(M)$ of $M$. Note that $M$ is ...
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Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing $$\langle -,-\rangle:M \otimes_R N \to R$$ is non-degenerate if, for all $n \... 0 votes 0 answers 73 views ###$\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones I am trying to refine my understanding of derived categories. Let$\text{Mod}_R$and$\text{Mod}^f_R$be respectively the categories of modules and finitely generated modules over a Notherian ring$R$... 1 vote 0 answers 35 views ### A bi-variate polynomial interpolation question Let$R$be a commutative unital ring, and$R^{m\times k}$denote the set of$m\times k$matrices with entries from$R$. A matrix$U\in R^{m\times m}$is elementary if$U$is obtained from the identity ... 1 vote 0 answers 52 views ### Prove that$f(M)=f^2(M)$implies$f(M)$is a direct summand of$M$whenever$\text{End}_R(M)$is a reduced ring Let$M$be a right$R$-module with the property that every homomorphism$\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to$S\to M$. If$S$has the property$f^2=0$implies$f=0$for every$f\in S$... 4 votes 1 answer 152 views ### Absolutely irreducible representation and splitting field Let$A$be a finite-dimensional algebra over a field$F$. A representation$M$of$A$is called absolutely irreducible if$M\otimes_FE$is irreducible as a representation of$A\otimes_FE$for all ... 7 votes 0 answers 232 views ### Split epimorphism of modules - does the finite case imply the infinite case? Let$k$be a field,$A$a finite dimensional$k$-algebra,$X$a finite dimensional indecomposable (left)$A$-module and$M$an infinite dimensional (left)$A$-module. Suppose further we have an ... 1 vote 1 answer 64 views ### Epimorphism going out of an inverse limit into a finite dimensional module Let$k$be a field and$A$a finite dimensional$k$-algebra. Given a sequence of inclusions$M_1 \subseteq M_2 \subseteq \dots$of$A$-modules consider the direct limit$M:= \bigcup_{i=1}^\infty M_i$. ... 1 vote 0 answers 57 views ### Is it possible to describe a$k$-basis for$M\otimes_{kH}N$when$M$is a$k[G\times H]$-module and$N$is a$k[H\times K]$-module? Suppose$k$is a field for Let$M$be a finitely-generated a$k[G\times H]$-module and let$N$be a finitely-generated$k[H\times K]$-module. Then in particular,$M$and$N$are finite-dimensional$k$-... 1 vote 0 answers 54 views ### When some idempotent ideals belong to the Gabriel filter of ideals for a hereditary torsion theory Let$\mathscr{I}_\sigma$be the Gabriel filter of ideals for a hereditary torsion theory$\sigma$over a commutative ring$R$. I am looking for equivalent conditions on either$\sigma$or$R$under ... 2 votes 0 answers 97 views ### modules over principal ideal rings Let$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$-modules? (i.e. a generalization of the theory ... 2 votes 1 answer 101 views ### Structure theorem for finitely generated$\Lambda$-modules - uniqueness part In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring$\Lambda = \mathbf{Z}_p[[T]]$. If$M$is a finitely generated torsion ... 3 votes 1 answer 181 views ### Split monomorphisms of modules - does the finite case imply the infinite case? Let$k$be a field,$A$a finite dimensional$k$-algebra,$X$a finite dimensional indecomposable (left)$A$-module and$M$an infinite dimensional (left)$A$-module. Further$X\subseteq M$and for ... 1 vote 0 answers 74 views ### Exterior algebra of free modules over Hopf algebras Let$H$be a commutative, cocommutative Hopf algebra over a field$\mathbb{K}$, and$M$a free Hopf module over$H$. Is the exterior algebra$\Lambda^k_\mathbb{K} M$with the diagonal$H$-action $$h \... 2 votes 1 answer 145 views ### On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev" I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ... 8 votes 1 answer 399 views ### For every ring R, is there a block-diagonal canonical form for a square matrix over R? This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring R. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily ... 7 votes 2 answers 107 views ### Rings of finite uniserial type If R is a ring and M an R-module, M is uniserial if its lattice of submodules is a chain. Over an Artinian R, the chain will be finite. From what I understand, deciding when two uniserial ... 5 votes 0 answers 78 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 ... 5 votes 0 answers 204 views ### Group cohomology of \mathbb{Z} vs \mathbb{Z}_p Let M be a continuous representation of \mathbb{Z}_p over \mathbb{F}_p, likely infinite-dimensional. There is the inflation map of group cohomology H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ... 8 votes 0 answers 245 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: ... 2 votes 0 answers 93 views ### The "matrix direct sum" monoid modulo unitary equivalence Given a commutative *-ring (R,*), let M(R,*) be the monoid whose elements are matrices over R of all possible shapes and entries, including those that have 0 columns or 0 rows. Let the ... 2 votes 0 answers 100 views ### Is this concept of a left-abelian category studied? A category is abelian if it is preadditive and it has a zero object, it has all binary biproducts, it has all kernels and cokernels, and all monomorphisms and epimorphisms are normal. Now we ... 1 vote 0 answers 66 views ### When is a bounded complex of RG-modules contractible? If we have a p-modular system (K, \mathcal{O},k), let R = k or \mathcal{O} and G a finite group. When is a bounded complex of RG-RG-bimodules \Gamma contractible? I've seen this response ... 2 votes 1 answer 128 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 ... 1 vote 0 answers 37 views ### A "spectral theorem" to SVD reduction for every commutative *-ring Given any commutative *-ring R of uneven characteristic, is it true that for every square matrix M and unitary matrix W, if W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W is ... 5 votes 1 answer 208 views ### Concept of an exact ideal of a module category Let R be a ring and \text{Mod}\,R the category of (left) R-modules. Consider an ideal \mathcal{I} of \text{Mod}\,R. For R-modules X and Y let \mathcal{I}(X,Y) be the collection of ... 21 votes 1 answer 2k views ### Reference request: a tale of two mathematicians I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so: When P. Gabriel presented the theorem in a conference [sometime around ... 3 votes 1 answer 220 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 ... 4 votes 0 answers 287 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 ... 2 votes 2 answers 153 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-... 1 vote 0 answers 77 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, ... 1 vote 0 answers 92 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 ... 4 votes 1 answer 287 views ### \|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\| when t=t^* Let A be a C^*-algebra, E be a (right) Hilbert A-module and t \in \mathcal{L}_A(E) be an adjointable operator satisfying t=t^*. Is it true that$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\... 3 votes 1 answer 176 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 ... 3 votes 1 answer 101 views ### On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ... 4 votes 1 answer 187 views ### Infinite linearly independent set in finitely generated module Let$R$be a (commutative, otherwise the answer is easy, see the comment below) ring and let$M$be a finitely generated$R$-module. Is it possible that$M$admits an infinite linearly independent set?... 0 votes 0 answers 49 views ### Tensor products and intersections of modules Is it true that$A\otimes_{\Bbbk} B \cap B \otimes_{\Bbbk} A = B\otimes_{\Bbbk} B$if$B \subset A$are$\Bbbk$-modules over a ring$\Bbbk$?. It works for$\Bbbk$a field. Does it work in any other ... 2 votes 0 answers 91 views ### Is there a category of "chains of modules" that behaves well with taking direct limits? I came up with the following definition of a category of certain "chains of modules" and want to know if this concept is already known and studied. Let$R$be ring. An object in our category ... 2 votes 1 answer 83 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 ... 4 votes 1 answer 160 views ### Is there essentially unique notion of module over monoidal stable$\infty$-categories? There is this (folklore?) fact: for a commutative ring$R$, the category of$R$-modules is equivalent to the category of internal abelian groups in the slice category$\operatorname{Commutative rings}/...
Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers \$n_1<n_2&...