Questions tagged [ring-theory]
This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.
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Proving a ideal of a ring
I am having trouble proving a.
My attempt:
I first have to prove that $0 \in \phi^{-1}(J')$. So $J'$ is an ideal, per definition $0 \in J'$. So take $x \in R$ such that $\phi(x)=0$ in $J'$. So $0 \in \...
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A problem from the book, "Topics in Algebra" by I.N Herstein from Chapter- 3 (Ring Theory) , Section-3.5, Page number- 140 (2nd Edition)
Let $R$ be a ring such that the only right ideals of $R$ are $(0)$ and $R.$ Prove that either $R$ is a division ring or that $R$ is a ring with a prime number of elements in which $ab = 0$ for every $...
- 1,885
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1
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Let R be a ring with unit element, R not necessarily commutative, such that the only right-ideals of R are (0) and R. Prove that R is a division ring.
Let $R$ be a ring with unit element, $R$ not necessarily commutative, such
that the only right-ideals of R are $(0)$ and $R.$ Prove that R is a division ring.
This was a problem from the book, "...
- 1,885
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Why $y$ is a rational integer?
I was reading that solution of $\mathbb{Z}[\sqrt{−n}]$ is not a UFD here Conclude that $\mathbb{Z}[\sqrt{−n}]$ is not a UFD.
But I do not understand this line "So either $d = 1, 2$ or $\gcd(y,\...
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1
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Is there a field (or field-like structure) that is distributive over both operations?
Is there a field (or field-like structure) that is distributive over both operations?
i.e. $(F,+,*)$ such that $\forall a,b,c \in F $
$a+(b*c) = (a+b)*(a+c) $
$a*(b+c) = (a*b)+(a*c)$
P.S.: For ...
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21
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Ideals are coprime if their product equals their intersection [duplicate]
I know that $I_{1}$ and $I_{2}$ are coprime indicates that $I_{1}\cap I_{2} = I_{1}I_{2}$. However, how do we show that $I_{1}$ and $I_{2}$ are coprime given that $I_{1}\cap I_{2} = I_{1}I_{2}$?
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3
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66
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Proof Explanation: Let $R$ be the ring of $2 × 2$ matrices with rational entries. Prove that the only ideals of $R$ are $(0)$ and $R.$
Let $R$ be the ring of $2 × 2$ matrices with rational entries. Prove that the only ideals of $R$ are $(0)$ and $R.$
This was a question from the book, "Topics in Algebra " by I.N Herstein in ...
- 1,885
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1
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37
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Commutativity of a ring with a ring homomorphism to itself $f(x) = x^2$
The full question is about a bit more than just commutativity, but I'm stuck on the commutativity part right now:
Let $R$ be a ring with the property that $f : R \rightarrow R, f(x) = x^2$, is a ring ...
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A question about tensor products of fin. gen. proj. modules and module maps
Consider non-zero bimodules $M,N$, and $P$, over a ring $R$, that ar additionally assumed to be finitely-generated and projective as left $R$-modules. Take a bimodule map $f:N \to P$, does it hold ...
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1
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Why does polynomial division carry from $\Bbb Z[x]$ to $\Bbb Q[x]$? [duplicate]
I'm reading Artin's Algebra. In 11.3.24, it is stated
Let $f$ be a monic integer polynomial, and let $g$ be another integer polynomial. If $f$ divides $g$ in $\Bbb Q[x]$, then $f$ divides $g$ in $\...
- 2,783
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Showing that $k[x]$ is integral over $k[x^2-1]$
For context I am working on Atiyah-Macdonald 5.4.
I want to show that the extension $k[x^2-1]\subset k[x]$ is integral. I believe this is the case using that $k[x^2-1]=k[x^2]$, which I believe can be ...
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Error in Herstein's "Topics in Algebra", zero ring has characteristic $1$ which is not prime
I was trying to prove the following statement:
"If an integral domain has a finite characteristic then the characteristic of the integral domain is a prime number"
This made me look at the ...
- 1,885
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49
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What is the definition of coset for an $n^{th}$ power element $\bar{y}$?
Here is the question I am trying to understand the solution of the second part in it:
Let $R = \mathbb Q[x,y,z]$ and let bars denote passage to $ \mathbb Q[x,y,z] / (xy - z^2).$ Prove that $\overline{...
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73
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Yes/No If $R$ be a ring with unit such that $a^2=a$ for $a \in R.$ Then$ R=\{0,1\}$ [duplicate]
Is the following statement true/false ?
If $R$ be a ring with unit such that $a^2=a$ for $a \in R.$ Then $ R=\{0,1\}$
My attempt : I think this statement is true
$(a+a)^2=a +a$
$(a+a)(a+a)=a+a \...
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Why showing that $\bar{xy} \in \bar{P}^{2}$ but that no power of $\bar{y}$ lies in $\bar{P}^2$ shows that it is a prime ideal?
Here is the question I am trying to understand its solution:
Let $R = \mathbb Q[x,y,z]$ and let bars denote passage to $ \mathbb Q[x,y,z] / (xy - z^2).$ Prove that $\overline{P} = (\bar{x}, \bar{z})$ ...
- 1,767
3
votes
1
answer
55
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Irreducible module over upper triangular matrices
Let $R$ be a ring and $M$ be an irreducible $R$ module. I want to create an irreducible $A$-module, where
$$A = \begin{pmatrix}R & R\\0 & R\end{pmatrix}.$$
I defined $\bar{M}$ as
$$\bar{M} = \{...
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2
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183
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Product ideals are the kernel of what ring homomorphism?
As I learned here Connections between the different characterizations of ideals? (Dedekind's ideal numbers, quotientable subsets, kernels), since the very beginning of humanity/Kummer's ...
- 8,148
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16
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Endomorphisms over direct sums and the proof of Artin-Wedderburn theorem
Let $A$ be a semi-simple ring. Then, the right $A$ module is isomorphic to a direct sum of minimal right ideals:
$$A_A\cong \bigoplus_{i=1}^m I_i^{n_i}, $$
where the $I_i$ are mutually non-isomorphic ...
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91
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Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
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52
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Surjective ring homomorphism from $\mathbb{Z}[\sqrt{-5}]$ to $\mathbb Z/3\mathbb Z$.
Question:
Let $R=\mathbb{Z}[\sqrt{-5}]$ and let
$$
\phi: \mathbb{Z}[\sqrt{-5}] \rightarrow \mathbb{Z} / 3 \mathbb{Z}, \quad a+b \sqrt{-5} \mapsto \overline{a+b} \quad(a, b \in \mathbb{Z}) .
$$
(a) ...
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25
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Isomorphism between quotients of two variable formal power series ring
$k$ is a field whose characteristic is not $2$, and $f(x,y)=x^2-y^2, g(x,y)=x^2+x^3-y^2$.
Exercise. Show that $k[[x,y]]/(f)\simeq k[[x,y]]/(g)$.
So far, I've shown the following.
$k[[x,y]]$ is an ...
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32
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Prime ideals equivalence
I'm reading Herstein's Non-commutative Rings book. His definition of prime ring is
A ring $R$ is said to be a prime ring if $aAb=(0)$ then either $a=0$ or $b=0$.
Then he presents the following Lemma:...
- 394
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1
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76
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Freeness of the algebra of formal power series?
Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. Then the algebra of formal power series $A[[x]]$ can be viewed as a $\mathbb{k}[[x]]$-module in a natural way. My question is ...
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Some possible typos in solution of 9.1.12 Dummit & Foote.
Here is the question I was trying to solve:
Let $R = \mathbb Q[x,y,z]$ and let bars denote passage to $ \mathbb Q[x,y,z] / (xy - z^2).$ Prove that $\overline{P} = (\overline{x}, \overline{z})$ is a ...
- 1,767
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Facts about Weyl algebra
I am trying to prove a few things about the first Weyl algebra, $W = k[x,y]/(xy-yx-1)$ over an algebraically closed field $k$ with $char(k)=p>0$. In particular, I am interested in nilpotent ...
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42
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Epimorphism and Surjective Homomorphsim [duplicate]
Here is a question I came across recently:
If a morphsim in Grp (category of groups) is an epimorphism, then it is a surjective group homomorphsim.
I believe it boils down to show that for any group $...
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1
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124
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Generalization of Eisenstein's Criterion [duplicate]
Let $f(X)=a_{2n+1}X^{2n+1}+\ldots+a_0\in \mathbb{Z}[X]$ with
$$\begin{align*}
a_{2n+1}&\not \equiv 0 \pmod p\\
a_{2n},\ldots,a_{n+1} &\equiv 0 \pmod p\\
a_n,\ldots,a_0&\equiv 0 \pmod{p^2} ...
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1
answer
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What $\mathbb Z/\ker f$ mean if $f$ is injective?
Here, I am trying to really understand the concept of cosets. From my understanding, given groups A and B, then A/B would mean the set of all the cosets of B in A. In that case, if I have $\mathbb Z/\...
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36
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Generators of $I$ in $\mathbb{Z}[X,Y]$
Let $p$ be prime and $(a,b)\in \mathbb{Z}^2$. Prove $I=\{f(X,Y)| f(a,b)\equiv 0 \mod p\}$ can be generated by three explicit elements.
Hilbert's Theorem tells me that $I$ must really be finitely ...
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1
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91
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How to prove that $IJ\subseteq P\implies I\subseteq P \vee J \subseteq P$
I'm having some trouble with the following exercise:
Let $R$ be a non trivial ring with unity, and let $P\neq 0$ be an ideal of $R$. Prove that if $R/P$ is a prime ring, then for all right Ideals $I,...
- 4,261
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1
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43
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Difference between $R/P$ and $R_P$
Let $R$ be a commutative ring and $P$ be a prime ideal.
Consider the quotient Ring $R/P$ and localization $R_P$, in the Borcherd's lecture, he said $R/P$ making $P$ minimal, but $R_P$ making $P$ ...
- 6,326
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Enveloping algebra of an algebra essentially of finite type (Weibel 9.4.5)
Let $k$ be a commutative Noetherian ring and $R$ be an algebra essentially of finite type (that is, $R$ is a commutative $k$-algebra and it is a localization of a finitely generated $k$-algebra). In ...
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1
answer
65
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Affineness of the algebra of formal power series?
Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. It is clear that when $A$ is an affine $\mathbb{k}$-algebra (that is, $A$ is finitely generated as a $\mathbb{k}$-algebra), ...
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2
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51
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The uniqueness of the identity elements are a request in field axioms? [duplicate]
I'm studying analysis and notice a conflict when some authors write about them. Sometimes is uniqueness of the neutral elements are in the axioms, sometimes is a corollary of the axioms which talk ...
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How to verify that the modulo map is a ring homomorphism?
The map f from $$\mathbb{Z} \to \mathbb{Z}/3\mathbb{Z}$$ that maps every integer to its modulo with 3 is a ring homomorphism. However, I'm having trouble verifying this as if we apply this map to the ...
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Length of a quotient of $1$-dimensional Noetherian local ring by a regular element
Let $(R,m)$ be a $1$-dimensional Noetherian local ring. $x$ is a nonzero divisor of $R$.
How to deduce the following identity
$$l(R/(x))=\sum_{P\text{ a minimal prime ideal}}l(R_P)l(R/((x)+P))\ ?$$
...
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1
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31
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One sided ideals of a semisimple ring.
Let $A$ be a semisimple ring. I'm wondering whether all ideals of $A$ are two sided. I know that all semisimple rings are both left and right semisimple. And, since $A$ is a semisimple module over ...
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Prove that $\mathcal{O}$ is a Euclidean Domain.
Here is the question I am trying to solve (Dummit & Foote, 3rd edition, Chapter 8, section 1, #8(a)):
Let $F = \mathbb Q(\sqrt{D})$ be a quadratic field with associated quadratic integer ring $\...
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Is this solution to show that the Ideal given by the kernel of$ f \in R \mapsto f(0,0) \in \mathbb C[X,Y]$ is not finitely generated correct?
Let $R \subseteq \mathbb C[X,Y]$ be the subring of all polynomials $f \in \mathbb C[X,Y]$ that can be written as $f = g(X)+X ·h(X,Y)$
a. Let $I \subset R$ be the kernel of the evaluation map $R \...
- 2,696
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+50
Automorphisms for direct products of finite commutative nilpotent rings.
Let $(R, +, \cdot)$ be an associative commutative nilpotent ring of cardinality $2^n$ such that
$$
r^2 = 0,
$$
for every $r\in R$. Also $(V, +)$ is a vector space over $\mathbb{F}_2$. Let $\...
- 1,324
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1
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88
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If $ab+ba=1$ and $a^3=a$ in a ring, show that $a^2=1$.
I've been stuck on this exercise for a while now. It asks:
If $ab+ba=1$ and $a^3=a$ in a ring, show that $a^2=1$.
I have a feeling this has to do with binomial expansion for rings, and the only ...
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1
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68
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The Ideal given by the kernel of map $ f \in R \mapsto f(0,0) \in \mathbb C[X,Y]$ is not finitely generated for this particular polynomial subring
In the following problem, $R$-module means left $R$-module and $R$ is a ring.
I have already proven these facts that may or not be needed:
-Show that an $R$-module $ M$ is finitely generated if and ...
- 2,696
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0
answers
81
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Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
- 375
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votes
1
answer
37
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Maximum height of radical of an ideal in a Noetherian ring
I am self-studying commutative algebra and basic algebraic geometry. At the time of solving some problems, I got stuck on the following.
Let $R$ be a Noetherian ring. $\mathcal{I}$ be an ideal of $R$ ...
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0
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36
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A commutative ring is a field if all proper ideals are prime [duplicate]
Let $R$ be a commutative ring with unity such that every ideal $I \subsetneq R$ is prime
Show that $R$ is an integral domain
Show that $R$ is a field (Hint: Given a non-zero element $a\in\mathbb{R}$, ...
1
vote
1
answer
57
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what is the difference between Theorem 15 and Problem 7.5.2 in Dummit & Foote
Here is Problem 7.5.2 in Dummit & Foote:
Let $R$ be an integral domain and let $D$ be a nonempty subset of $R$ that is closed under multiplication. Prove that the ring of fractions $D^{-1} R$ is ...
- 1,767
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0
answers
38
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Understanding the uniqueness of the ring of fraction proof in D&F.
Here is the part that I do not understand in the proof of theorem 15 (on pg. 261 of Dummit & Foote, 3rd edition):
Here is Theorem 15:
Can someone tell me please how this shows the uniqueness of ...
- 1,767
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1
answer
52
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How do I complete the proof of the composition rule for the functor that sends a ring $R$ to its group of units $R^×$?
I have to prove that there is a functor Ring $\rightarrow$ Grp that sends a ring $R$ to its group of units $R^×$. In order to to that :
Let $R,S,T$ be rings, $h$ a homo of rings from $R$ to $S$ and I ...
- 27
1
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1
answer
60
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Start of the proof that there is a functor Ring $\rightarrow$ Grp that sends a ring $R$ to its group of units $R^×$
I have to prove that there is a functor Ring $\rightarrow$ Grp that sends a ring $R$ to its group of units $R^×$. In order to to that I would start like this:
Let $R,S,T$ be rings, $h$ a homo of ...
- 27
4
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0
answers
64
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The stable category of $\mathbb{Z}$
Is there an alternative description/characterization of the stable module category of Abelian groups? I guess that the category of torsion groups is a subcategory of it, but is it all of it?
What is ...
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