All Questions
Tagged with modules ra.rings-and-algebras
195 questions
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31
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Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 923
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0
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91
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Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
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votes
0
answers
76
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Does a matrix ring over a ring satisfy the Koethe conjecture if the coefficient ring itself satisfies the Koethe conjecture?
I just want to know whether the following statement is true or false.
If $R$ is a ring satisfying the Koethe conjecture, then the matrix ring over $R$ also satisfies the Koethe conjecture.
Or is it ...
1
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0
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58
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When is a bimodule that is projective as a right and as a left module also projective as a bimodule
Are there practical criteria for determining when a bimodule that is projective as a right and as a left module is projective as a bimodule? Some illustrative examples of what goes wrong and what goes ...
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59
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Bimodule endomorphisms of a bimodule over a noncommutative ring
Let $R$ be a noncommutative ring and $M$ an $R$-$R$-bimodule that is projective as left $R$-module. We know that the bimodule of left $R$-module endomorphisms ${}_REnd(M)$ is isomorphic to the tensor ...
2
votes
1
answer
210
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Direct product of direct sum of a flat module
In the book "Rings and Categories of Modules" by Anderson & Fuller, this problem is given: If $V^A$, i.e. the direct product of the module $V$ by the index set $A$, is flat for all sets $...
- 355
1
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1
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87
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An example of a commutative ring which is not SIP
Recall that a module $M_R$ ($R$ is a unital ring) is called an SIP-module if the intersection of any two summands of $M$ is a summand. The ring $R$ is called (left) right SIP-ring if the module (${}...
- 324
1
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1
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75
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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_{...
- 1,644
3
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2
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468
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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 ...
- 823
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2
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214
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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?
- 139
2
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2
answers
416
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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? ...
- 319
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60
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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 \...
- 223
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71
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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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46
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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}\...
- 620
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3
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434
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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)$ ...
- 1,066
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1
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88
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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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198
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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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242
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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}$ ...
- 307
1
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1
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84
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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} ...
- 324
8
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1
answer
191
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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 ...
- 63.9k
5
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302
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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. ...
- 985
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1
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214
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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_{...
- 321
11
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1
answer
159
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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 ...
- 301
1
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1
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130
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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.
...
- 324
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76
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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=\...
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59
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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 ...
- 4,943
4
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1
answer
186
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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 ...
- 1,418
3
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225
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Intersection of two modules (and sub-modules) under tensors
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 (...
- 31
4
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1
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142
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Property of simplicity and semi-simplicity under base change of base field
Suppose $K$ is a field of characteristic $0$ and $A$ is a $K$-algebra. Let $F$ be a field extension of $K$ and let $M$ be an $A$-module. What can we say about simplicity or semi-simplicity of $A_F$-...
5
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1
answer
256
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Classifying indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$
I'm now interested in classifying the indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ : the group ring of $\mathbb{Z}/p\mathbb{Z}$ over the ring $\mathbb{Z}/p^{2}\...
- 1,013
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2
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387
views
Torsion of modules
Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
- 2,390
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0
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102
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When do two path algebras share an underlying graph?
Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction.
Since ...
- 433
3
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0
answers
82
views
Example of a secondary representation of a module that is not a direct sum
Let $A$ be a commutative ring. An $A$-module $M$ is said to be
secondary if $M\neq 0$ and for each $a\in A $, the endomorphism
$\phi_a:M\to M$ defined by $\phi_a(m)=am$ for $m\in M$ is either
...
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3
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0
answers
128
views
Which definitions of "local module" have gotten traction?
It seems like "local module" has been defined a lot of ways:
if 𝑀 has a largest proper submodule. (This math.se post)
if it is hollow and has a unique maximal submodule (Singh, Surjeet, ...
- 1,507
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0
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173
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Where could a paper on a unification of matrix decompositions be published?
I've got a paper which shows that when the spectral theorem (as a statement that every self-adjoint matrix can be unitarily diagonalised) is naively generalised to $*$-algebras other than the complex ...
- 4,943
2
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1
answer
89
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Primitive group rings and endomorphism rings
It is known that, for any group $G$, there exists a group $H$ containing $G$ such that the group ring $F[H]$ for some field $F$ is primitive, see Formanek, Edward; Snider, Robert L., Primitive group ...
user491484
28
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2
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863
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$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
- 18.7k
2
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1
answer
194
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Structure of reflexive modules over regular local rings
Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in Reflexive modules over a 2-dimensional ...
- 429
1
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1
answer
118
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Subrings, submodules, and flatness
Let $R$ be a ring, and $S$ a subring of $R$. Let $M$ be a right $R$-module, and $N$ a right $S$-submodule of $M$. If $N$ is flat (or faithfully flat) as a right $S$-module, does it then follow that ...
- 1,144
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1
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284
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Does the choice of the algebraically closed field of characteristic $p$ have influence on the module category?
Let $G$ be a finite group and $p$ be a prime number dividing $|G|$.
Let $k$ be the algebraic closure of $\mathbb{F}_p$.
Let $K$ be another algebraically closed field of characteristic $p$ which is not ...
- 237
3
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1
answer
171
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Modules with special properties
$\DeclareMathOperator\End{End}$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 $\...
- 26.5k
3
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0
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104
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Are there (co-)homological obstructions to the extendability of a homomorphism?
Let $k$ be a commutative ring and $A \subset B$ an extension of $k$-algebras. Can we associate to this extension a (co-)chain complex so that its (co-)homology $H$ allows for statement such as
"...
- 203
4
votes
1
answer
564
views
Nondegenerate pairings versus perfect pairings for finitely generated projective modules
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 \...
- 145
1
vote
0
answers
66
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 $f\in S$...
- 111
5
votes
1
answer
707
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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 ...
- 951
1
vote
1
answer
111
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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 $M:= \bigcup_{i=1}^\infty M_i$. ...
- 696
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
1
answer
520
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 ...
- 4,943
7
votes
2
answers
166
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 ...
- 549
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 ...
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