Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Can a zerodivisor reduce both the depth and the dimension?
In this question $R$ is a commutative noetherian local ring with unity.
One can construct examples of rings $R$ and zerodivisors $z$ such that $\dim R/(z)=\dim R-1$, e.g., $S\colon=k[a,b,c],\ \...
- 3,101
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171
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Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group
Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...
- 622
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40
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Hilbert functions of graded modules generated by mapped generators
I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...
- 11
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335
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Which sheaves on a projective bundle are flat over the base scheme?
Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...
- 1,025
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1
answer
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A Module with $Ext^i(M,R) = 0$ for all $i > 0$
Let $M$ be a finitely generated module over a noetherian local ring $R$. We can take our ring to be Cohen-Macaulay. Suppose $M$ satisfies the condition $Ext^i(M,R) = 0$ for all $i > 0$. We want to ...
- 200
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1
answer
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When are intersections of finitely generated field extensions finitely generated?
Let $k$ be a field, and let $E$ and $F$ be fields extending $k$, both contained in some single extension of $k$. If $E$ and $F$ are finitely generated (as fields) over $k$, must $E\cap F$ also be ...
- 6,199
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Serre condition $(S_n)$
We know that a finitely generated $R$-module $M$ satisfies the $(S_n)$ condition if $$\operatorname{depth} M_p \geq \min(n,\dim M_p)$$ for every $p\in \operatorname{Spec}R$.
It's well known that ...
- 418
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3
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2k
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Chevalley's valuation extension theorem and the axiom of choice
Hello,
Do we know if the axiom of choice is needed for Chevalley's valuation/place extension theorem (i.e. the theorem that states that for every valued field and a field extension, one can extend ...
- 43
1
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116
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Prime ideal ramified in extension if and only if certain polynomial divides another one?
Let $k$ be a field of characteristic $ \neq 2$, and let $f \in k[T]$ be a polynomial of degree $\ge 1$ which is square free. Let $K$ be the quadratic extension $k(T)(\sqrt{f})$ of $k(T)$. I know that ...
- 369
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Kaplansky's theorem for graded local rings
Hello!
This is a very short question:
Given a local graded Noetherian ring $R_{\bullet}$, is it true that any graded projective module over $R_{\bullet}$ is free?
In the ungraded case, this is true,...
- 2,756
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86
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$\mathbb Z$-torsion for some quadratically presented Lie rings
$\newcommand{\Z}{\mathbb{Z}}$
I asked this question on MSE but no answer so far, so I'm also asking it here.
Let $L$ be a Lie ring (a Lie algebra over $\Z$) with generators $x_1,\dots,x_n$ and ...
- 8,524
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4
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444
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Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$
Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x_1, \dotsc, x_n] \to A$. For $f \in A$, define $\...
- 7,338
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430
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Normality condition on graded algebra
Let $\mathbb G_a$ denote the additive group of complex numbers.
Definition:
Let $V \subset Y$ be a dense open subset of the affine variety $Y$ and
$\pi : P \longrightarrow V$ a $\mathbb G_a$-...
- 337
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2
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377
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Is there an intuitionistic generalized boolean algebra (of Stone)?
A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...
- 889
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255
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Socle of Almost Complete Intersections
Let $(A,m)$ be a complete Artinian local ring over a field $K$.
We focus on almost complete intersection ring $A$ of the form
$A = K[[X_1,...,X_N]]/(f_1,...,f_{N+1})$.
We assume that none of $f_i$ ...
- 781
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139
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A question about prime elements in a specific integral domain
Let $R,\Omega$ be two integral domains such that $R$ is Noetherian and $\Omega=R[\alpha]$ for some $\alpha\in \Omega$.
If there are infinite prime elements in $R$, can we proof that there are ...
- 1,997
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1
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182
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When do all annihilators of primitive idempotents intersect in {0}?
maybe this is silly but:
for which class of rings (or commutative rings) R may I write
An element a of R is zero iff
for every primitive idempotent e, ea is zero
?
That is, primitive idempotents "...
4
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1
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454
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Example of an overring of an integral domain which is not a ring of quotients?
Hi!
I'm trying to make headway on a question for my undergraduate honors thesis, specifically the question of which rings of integer-valued polynomials if any satisfy the QR-property; that is, the ...
- 147
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269
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Non-commutative normalization
Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that
a) $Z$ is a noetherian domain
b) $A$ is a finitely generated module over $Z$.
...
- 7,037
1
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2
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388
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quotient of integral polynomials not being integral
So let $R$ be an integral domain and $K$ its fraction field. Let $f,g,h\in K[x]$
be 3 monic polynomials such that $f=gh$. So I would like to have a simple example
of a ring $R$ for which one has that $...
- 7,609
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0
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134
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Integral basis of an extension of complete fields
Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$.
Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure $\...
- 355
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1
answer
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Number of graphs with a given number of nodes, edges and triangles
Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and triangles?...
- 902
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296
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A roadmap to learn about finite-dimensional commutative associative real or complex unital algebras
I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the ...
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1
answer
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Irreducibility of some trinomials modulo $p$
Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in
http://www.mscand.dk/article.php?id=1472,
(If the link does not work try googling: ...
- 1,641
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0
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107
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Question about GIT: when is the map $\pi:X//_\theta G\rightarrow X//G$ birational?
I want to ask somebody who is more familiar with the theory of GIT quotients than I am, if there is a nice list of conditions on the action of a reductive group $G$ on an affine variety $X$ over ...
- 790
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171
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A characterization for the ideals of $A+XB[X]$ and $A+XB[[X]]$
Let $A \subseteq B$ be an extension of commutative rings with identity. Then $A+XB[X]$ and $A+XB[[X]]$ are the polynomial and power series rings over $B$ whose constant terms are in $A$. Is there any ...
- 11
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177
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When is a commutative ring the limit of its factor rings?
Let $R$ be a commutative ring. Consider the limit of rings $L = lim_{I \in Spec(R)}(R/I)$. Then there is a canonical map $R \to L$. The question is when this map is an isomorphism.
For example, this ...
- 4,459
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106
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Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
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votes
2
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2k
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Cardinality of maximal linearly independent subset
M a finitely generated module over a commutative ring A. I can't think of an example of two maximal linearly independent subsets of M having different cardinality. I know that they all have the same ...
- 2,857
5
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1
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861
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Smooth function algebra on cartesian product and beyond
Short question:
Let $M$ and $N$ be smooth manifold, with appropriate smooth function algebras
$C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$.
Can we express the smooth function algebra of ...
- 624
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0
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212
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Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$
This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will trivially ...
- 10.5k
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147
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Behaviour of an étale morphism under Galois action on points
Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such ...
- 181
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1k
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Tensor product of commutators vs. commutator in a tensor product
Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
- 4,015
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3
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818
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Is the restriction map an epimorphism of commutative rings?
Let $i : U \to X$ be a quasicompact open immersion of schemes. I would like to know whether the canonical morphism $i_* \mathcal{O}_U \otimes_{\mathcal{O}_X} i_* \mathcal{O}_U \to i_* \mathcal{O}_U$ ...
- 63.1k
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1
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675
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Resolution of a module as an $A_\infty$ module over resolution of an algebra
The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...
- 1,545
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1
answer
383
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Dimension of Ext modules [closed]
Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
- 21
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381
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Samuel multiplicity
Let $X$ be the hyper-surface defined by
$$f:=\sum_{i=1}^k x_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x_1^{n-1},\dots , x_k^{n-1}) $$
What is ...
- 2,384
2
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123
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a multiplicatively closed subset of $\mathbb{Z}[x]$
Consider the ring $R=\mathbb{Z}[x]$. Consider $S=\lbrace\text{ }f(x)=a_nx^n+\dots+a_0\in\mathbb{Z}[x]\text{ }|\text{ }a_i\geq 0\text{ or } a_i
\leq 0 \text{ for all } i\rbrace$. It is easy to see that ...
- 239
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135
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Monomial algebras and depth
Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence.
Assume $...
- 11
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0
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173
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When a ring is a polynomial ring?
In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k \...
- 271
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306
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Smooth affine algebraic subgroups as complete intersections
Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular ...
- 3,637
6
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1
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175
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Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?
Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ \...
- 3,245
7
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1
answer
772
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Maximal ideals of the rings of Baire-One Functions
A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all $x\...
- 1,788
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2
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2k
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Homogeneous ideal and its system of generators
Let $I$ be a homogeneous ideal in a graded commutative ring $R$, $S$ be its minimal system of generators.
What is the conclusion that we can say about the element in $S$ ? Is the cardinality of $S$ ...
- 325
1
vote
0
answers
81
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2 questions about "monogenic" coordinate rings of affine curves
Suppose $k$ is an algebraically closed field with characteristic 0 and let $X \subset \mathbb{A}^n$ be an irreducible curve.
If $f \colon X \to \mathbb{A}^1$ is finite of degree $d$, then the ...
- 243
1
vote
1
answer
787
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injectivity is a local property over noetherian rings [closed]
Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How I can show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$, then $M$ is ...
- 15
3
votes
2
answers
348
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Generating function for Poincare series of dimensions of $Tor^R_i(k,k)$
Let $R=k[t_1,\ldots,t_m]$ be a polynomial ring over a field $k$ and $I=(f_1,\ldots,f_r)$ an ideal of R. The $f_i$ shall be homogeneous for the natural grading of R and of degree greater than 1. Let $S=...
- 708
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votes
2
answers
1k
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Solve for $A$ and $B$ in $AXB=Y$
Let $R = \mathbb{Z}[x_{1}, \dots, x_{r}]$.
Let $X$ be $n \times n$ matrix with entries in $R$.
Let $Y$ be $m \times m$ matrix with entries in $R$ formed from $\mathbb{Z}$-linear or $\mathbb{R}$-linear ...
- 13.9k
-3
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1
answer
274
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Isomorphic quotient of a Module over Noetherian commutative algebra [closed]
I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
- 3
5
votes
2
answers
966
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Krull-Schmidt Analogue for Complete / Graded Rings
Over the ring $\mathbb{Z}$, all finitely-generated modules decompose uniquely as a direct sum of indecomposable submodules; that's the Krull-Schmidt theorem.
I'm given to understand that if a (...
- 146