All Questions
Tagged with quotient-group modules
16 questions
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Prove that $\frac{I\cap J}{IJ}\cong\frac{k[x,y]}{\langle xy\rangle}$ for $I=\langle x^2y \rangle$ and $J=\langle xy^2\rangle$
$R=k[x,y]$ is a $R$ module,where $k$ is a field. $I=\left<x^2y\right>$ and $J=\langle xy^2\rangle$. Prove that
$$\frac{I\cap J}{IJ}\cong\frac{k[x,y]}{\left<xy\right>}$$ as $R$ modules.
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For groups, rings, or $R$-modules $A, B, C$ with $C\subseteq B$, does $A/B \cong A/C$ imply that $B=C$?
I think this is true if the module is finite, using Lagrange's Theorem. However, is this the case for infinite modules (or rings, or groups)? This comes from me trying to prove that tensoring is right-...
- 1,588
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2
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Compute $T(M)$, where $M=\mathbb{Z}[x]/(x^2-9)$
Given an $R$-module $M$, let
$$T(M):=\{ x\in M \ | \ rx=0, \ \text{for some} \ r\ne 0 \ \text{in} \ R\}$$.
Let $R=M=\mathbb{Z}[x]/(x^2-9)$. Compute $T(M)$.
Here is what I am thinking, let $x=f(x)+(x^2-...
- 2,481
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170
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$R$-module isomorphisms $M/A\cong B$ And $M/B\cong A$
Let $R$ be a commutative ring, and let $M$ be a module with submodules $A,B \subset M$. Show that if $A \cap B=0$ and $A+B=M$, then there are $R$-module isomorphism $M/A \cong B$ and $M/B \cong A$.
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- 2,481
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Does $S\subset R$ imply $A\otimes_{R} A \subset A\otimes_{S} A$?
I'm currently trying to prove Proposition 3.2 from the paper "Galois correspondence for Hopf Bigalois Extensions" by Peter Schauenburg, focusing on the last two implications it declares as &...
- 3,609
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What is the group structure behind quark color charge?
Suppose you take the free $\mathbb Z$-module with two generators $\{P,N\}$ and quotient out by the relation $P+N = 0$. You get a one-dimensional module that has two additively independent elements $P$ ...
- 10.5k
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630
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What is the quotient of a quotient module? [closed]
Say you have $N=M\big/K$, where $M$ is an $R$-module and $K$ lies inside $M$. When I take $N\big/IN$, with $I$ being an ideal in $R$, is that equal to $$M\big/\big(IM + K)\ ?$$ How do you find that ...
- 996
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Number of subgroup $G<\Bbb Z^3$ such that $\Bbb Z^3/G\simeq \Bbb Z/3\Bbb Z\oplus\Bbb Z/3\Bbb Z$ [closed]
Compute the number of subgroups $G<\Bbb Z^3$ such that $\Bbb Z^3/G\simeq \Bbb Z/3\Bbb Z\oplus\Bbb Z/3\Bbb Z$.
Possible forms of $G$ are $G = \Bbb Z\oplus 3\Bbb Z\oplus 3\Bbb Z$, $3\Bbb Z\oplus 3\...
- 5,157
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154
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To prove $O/aO$ is finitely generated
Let $R$ be one dimensional Noetherian integral domain and $K$ be it's fraction field. Let $L/K$ be a finite extension, and $O$ be the integral closure of $R$ in $L$.
Then, why $O/aO$ ($a$ is an ...
- 6,648
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1
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91
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A natural way of thinking about quotient of groups
Given a ring $A$, the subsets by one could quotient it while maintaining the ring structure are precisely the ideals. This happens because when considering $A$ as a module over itself, the ideals as ...
- 720
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1
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183
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Colon ideal and Cyclic modules
I'm reading module theory as a beginner. The following problem might be super silly. I apologize for flooding SE with this kind of basic question.
Let $R$ be a ring with $1$, and $\mathscr{m}$ and $\...
- 3,522
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1
answer
143
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Making the module structure trivial
Let $G$ be a group (not necessarily abelian) acting on an abelian group $M$. Consider the group ring $\Bbb Z[G]$ (which is in fact an algebra). The action of $G$ on $M$ makes $M$ as a $\Bbb Z[G]$-...
- 1,920
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607
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Decomposition of an abelian group subject to certain relations
I've been trying to solve the following problem:
Suppose that $G$ is an abelian group, generated by $x_1, x_2, x_3, x_4$, and subject to the relations:
$$4x_1 - 2x_2 - 2x_3 = 0; 8x_1 - 12x_3 + 20x_4 =...
user437748
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165
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quotient module $\mathbb{Z}^3/(2,0,3)\mathbb{Z}$
Is $\mathbb{Z}^n/(m,a_2,...,a_2)\mathbb{Z}\cong\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}^{n-1}$ for $0<m<a_2,...,a_n\in\mathbb{Z}$ (as $\mathbb{Z}$-modules)? If not, how can you compute something ...
- 339
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1
answer
249
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Confusion about the quotient $\mathbb Z^4/H$
Let $f:\mathbb Z^3\to\mathbb Z^4$ be the group homomorphism given by $$f(a,b,c)=(a+b+c,a+3b+c,a+b+5c,4a+8b).$$ Let $H$ be the image of $f$. Find an element of infinite order in $\mathbb Z^4 /H$ and ...
- 12.1k
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quotient of direct sum of modules is the direct sum of the quotients?
I am making a proof and at a certain point I need to use the following, but I am not sure that it is true:
Let $M=M_1\oplus M_2$ be a module over a $K$-algebra $A$, with $M_1,M_2\leq M$. If $N\le M$, ...
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