Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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$U_1(\mathbb{Z}G)$ is a finitely generated FC-group.

If each member in support of an element in $\mathbb{Z}G$ is centralized by a subgroup of finite index in $G$, then why does it imply that $U_1(\mathbb{Z}G)$ is a finitely generated FC-group., where ...
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how to show that an ideal is convex [on hold]

I need to show that the ideal $J=(i)$ in $C(\mathbb R)$ where $i$ is the identity function, $C(\mathbb R)$ is the ring of all continuous functions on the real numbers, is a convex ideal.
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definition of Artin map

Let $L/K$ be an unramified Abelian extension. Then the Artin symbol $ \left ( (L/K) / \mathfrak p \right) $ is defined for all prime ideals $\mathfrak p$ of $\mathcal O_K$. (because in an Abelian ...
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When does each action corresponds to a homomorphism and an anti-homomorphism.

If I adopt for function evaluation and function composition the convention $f(x)$ and $(f\circ g)(x) = f(g(x))$, and if $G$ is a left group action on some set $X$, then to this left action their ...
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Should an object in the category always be a formal mathematical structure?

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
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If $A$ is a submodule of $B$ and $B$ is a submodule of $C$, is $A$ a submodule of $C$?

Let $C$ be a commutative ring (with 1, if this matters). If $A$ is a submodule of $B$ and $B$ is a submodule of $C$, is $A$ a submodule of $C$? I can't really prove that it is true because it is ...
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39 views

Problems with understanding the proof of noetherian ring

If $M$ is an $R$-module, the the following are equivalent: 1. M is finitely generated 2. M satisfies the ascending chain condition 3. Every non-empty set of submodules of M contains at least one ...
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$[U \cap H : G \cap H] \le [U :G]$

Is this always true that $[U \cap H : G \cap H] \le [U :G]$ , where U is a group and $G,H$ are subgroups of $U$? My trials for $\mathbb{Z}$ were giving affirmative answer but how to prove it, if it ...
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a question about abstract algebra,prove that the ring is commutative. [duplicate]

(1)A ring R is a booleean ring if for every $a\in R$,$a^2=a$. Show that every Boolean ring is a commutative ring. (2)Let R be a ring,where $a^3=a$ for all $a\in R$.Prove that R must be a commutative ...
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Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$ Does this proof work? Any criticism appreciated: We rewrite $\operatorname{lcm}(a,b) = ...
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The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
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Criterions for $U_1(\mathbb{Z}G)=G$ i.e. units to be trivial in $\mathbb{Z}G$

Notation- $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. $u=\sum_{g\in G} a(g)g \in U(\mathbb{Z}G)$ such that $\sum_{g \in G} a(g)=1$ 1) I have done theorem by ...
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1answer
41 views

Why $R/AB$ is cyclic?

Why $R/AB$ is cyclic when $A,B$ are ideals of $R$? I know a cyclic module is a module that can be written as $Rm$ and $Rm$ is isomorphic to $R/Ann(m)$. However, I can't see why the module is ...
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27 views

Factor into primes in Dedekind domains that are not UFD's?

Does it make sense to factor numbers into prime numbers in Dedekind domains that are not unique factorization domains? I can't really see how it would make sense.
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Element order proof [on hold]

$n\in \mathbb{Z}$ and $\overline{a}\in U(\mathbb{Z}_n)$ order is $kl$. Prove that $\overline{a}^k$ order is $l$. Any ideas on how to approach this? It seems to follow straight from power definition. ...
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Prove that $\alpha$ and $\beta$ are homomorphisms from $F(\mathbb{R})$ onto $\mathbb{R}$

Let $\alpha : F(\mathbb{R}) \to \mathbb{R}$ be defined by $\alpha(f)=f(1)$ and let $\beta : F(\mathbb{R}) \to \mathbb{R}$ be defined by $\beta(f)=f(2)$ Prove that $\alpha$ and $\beta$ are ...
2
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1answer
56 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all $2 × 2$ matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? We know, in the ring $\mathbb{Z}$, ...
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28 views

Commutators of a symmetric group

I am trying to prove that the commutator subgroup of $S_n$ ($S_n$ is a symmetric group on $[n]$), $[S_n,S_n]$ consists solely of commutators $s_1^{-1}s_2^{-1}s_1s_2$ for some $s_1,s_2\in S_n$. Any ...
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42 views

Whether the number $a$ is algebraic over $\mathbb{Q}$ [on hold]

Is the number $a = \displaystyle \left( \frac{(1 + \sqrt[3]{7})^{\tfrac{7}{5}}}{(\sqrt[3]{7} - 7)^3 + 77} \right)^{13}$ algebraic? If so, is algebraic degree of $a$ bounded by $15$?
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is the image of a group inside its profinite completion normal? characteristic?

Let $G$ be a finitely generated, residually finite group and $\widehat{G}$ its profinite completion. Must the natural image of $G$ inside $\widehat{G}$ be a normal subgroup of $\widehat{G}$? Must it ...
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What does a basis of an affine module correspond to in a torsor?

An affine module, if and only if the module is free, by definition has one or more bases. My understanding is that an affine module over the ring $\mathbb Z$ can be converted into an equivalent ...
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Bourbaki - Algebra Chapter IV - Section 6, Exercise 9(b)

Let $S_i(X_1,\dots,X_n)$ be the elementary symmetric functions in the variables $X_1,\dots,X_n$. Let $r_1,r_2,\dots,r_n$ be $n$ rational functions in the $X_1,\dots,X_n$. Let $T$ be a variable ...
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Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
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Bourbaki Algebra Chapter IV - Exercise 9(b) [duplicate]

I can't figure out to do this when at least one of the j's is less than n.
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2answers
47 views

R is commutative ring with identity & define $\circ$ on $R$ by for any $a,b \in R$ $a \circ b=a+b-ab$ Prove the following

Let R be a commutative ring with identity. Define a new operation $\circ$ on $R$ by for any $a,b \in R$ $$a \circ b=a+b-ab$$ a) Prove that $\circ$ is associative b) Prove that R is a field iff the ...
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Is $\sqrt[\beta]{\alpha}$ algebraic?

If $\alpha \in \mathbb{C},$ algebraic (over $\mathbb{Q}$) and $ \ \beta \in \mathbb{N} \ $then is $ \sqrt[\beta]{\alpha}$ algebraic? This is my attempt at a proof: Given that $\alpha$ is algebraic ...
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Show that $\bar{a}_{n}(\bar{x})^n+···+\bar{a}_{1}\bar{x}+\bar{a}_{0}=0_{F[x]/I}$

Let $F$ be a field, $f(x)$ be an irreducible polynomial in $F[x]$ and $I =(f(x))$. Let $f(x)= a_nx^n+···+a_1x+a_0, a_i \in F$ for $i=0,...,n$. And, $\bar{x} = x + I ∈ F[x]/I$ and $\bar{a_i} = a_i + I ...
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Kernel of a $R$-linear map is uniquely determined?

In my algebra textbook there is the following exercise: Let $M$ and $N$ be two left $R$-modules and $f:M\longrightarrow N$ a $R$-linear map. Show the kernel $L$ of $f$ is uniquely determined by the ...
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Smith Normal Form and quotient $\mathbb{Z}^{3}/M \mathbb{Z}^{3}$

I am learning modules and the Smith Normal Form, but I got stuck in the following: I found the Smith Normal Form of $$M = \begin{pmatrix} 21 & 0 & 1 \\ 8& 4 & 1\\ 3& 8 & 1 ...
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Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. [duplicate]

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. Ok, so I'm just looking for some confirmation that I'm doing this correctly. If we suppose $x,y \in R$ Let's ...
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Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
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Paradoxical Decomposition

A paradoxical decomposition of a group $G$ is a decomposition of $G$ into disjoint subsets: $G=U\cup S_1\cup S_2\cup\cdots\cup S_m\cup T_1\cup T_2\cup\cdots T_n$ so that there exist elements $g_1, ...
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understanding of Artin's proof of “$A_n$ is generated by $3$-cycles”

A quick proof for "$A_n$ is generated by $3$-cycles ($n\geq 4$)" is calculating the product of possible two $2$-cycles. I read the following different proof from Artin's Algebra(2nd): This is ...
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Algebraic and transcendental functions over $\mathbb Z_n$ — is this a known result?

I have proved a result that seems (to me) interesting, and I am wondering whether it is a known result, and if not whether it seems interesting to others. The result is as follows: Let $R=\mathbb Z ...
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If $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $ R $-modules, then $I + J = R$. [duplicate]

If $R$ is a commutative ring with identity and $I$ and $J$ are ideals of $R$ such that $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $R$-modules, then $I + J = R$. I know this is the ...
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Show that the group is isomorphic

If $H$ and $K$ are normal subgroups of $G$ and $H$ $\bigcap$ $K$ = {$e$}, prove that $G$ is isomorphic to a subgroup of $G/H$ $\bigoplus$ $G/K$. What I have so far: Let $\phi$: $G$ $\mapsto$ $G/H$ ...
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Show that infinite direct sum of injective modules is not an injective module [on hold]

I want an example to show that infinite direct sum of injective modules is not injective
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Is it true that if $g,h\in G$ have order $p$, where $p$ is prime, the only possible order of $\langle g\rangle \cap \langle h \rangle$ is $p$?

Is it true that if $g,h\in G$ have order $p$, where $p$ is prime, the only possible order of $\langle g\rangle \cap \langle h \rangle$ is $p$? Here is my attempted proof of the result. Is it ...
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How to multiply permutations together?

This is straight from an exam question: Find the order of the permutation $(1465732)(358)(79)$ in $S_9$ So I understand that I first have to write this permutation in disjoint cycle notation, but I'm ...
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Are ideals also rings?

I am learning about rings and ideals. But I am confused about something. My book (Gallian) says that an ideal of a ring by definition is a subring. But I have talked to other people who insist that an ...
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Show that if $R$ is commutative, then $\mathrm{Ann}(n_1) + \mathrm{Ann}(n_2) = R$.

Let $R$ be a commutative ring with $1$, and let $M$ be a cyclic $R$-module with generator $m$ (so that $M=Rm$). Suppose that $M = N_1 \bigoplus N_2$ for some submodules $N_1$ and $N_2$ of $M$. Let ...
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Example of a ring where all but two of its elements are units

One way of viewing a field is just as a ring where all but one of its elements (namely $0$) is a unit. I'm looking for rings (commutative with a 1) where all but two of its elements are units. I found ...
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Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
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Prove if a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$.

If a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$. I can prove that it is two sided, but I can't prove that it is ...
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An element in $\mathbb{Q}(c)$ where $c$ is a root.

Let $c$ be a root of $f(x) = x^3 + 4x^2 - 6x + 2.$ An element of $\mathbb{Q}(c)$ can be expressed uniquely in the form $a_2c^2 + a_1c + a_0$ for some $a_2,a_1,a_0 \in \mathbb{Q}$. If we express $c^4$ ...
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Why is the trace submodule of $FS_n$ the unique $1$-dimensional submodule?

Suppose $V$ is an $n$-dimensional vector space over a field $F$, with basis $e_1,\dots,e_n$. Then $S_n$ acts on $V$ by the action $\sigma\cdot e_i=e_{\sigma(i)}$, and $V$ is a $FS_n$-module, where ...
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Relation between reduced finite algebra, prime ideal and field extension

Is it true that if $L$ is a reduced finite dimensional commutative algebra over a field $K$ (which is finite or of characteristic $0$) and if $\mathfrak p$ is a prime ideal of $L$, then ...
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Prove $\phi$ is a homomorphism.

Let H and K be normal subgroups of a group $G$ with $H \subseteq K$. Define $\phi: G/H \rightarrow G/K$ by $\phi(Ha)=Ka$ Prove $\phi$ is a homomorphism. We are given a function $\phi$, to prove ...
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Prove that if $M$ is a simple left $R$-module, $Ann(m)$ is a maximal left ideal of $R$ and $M \cong R/Ann(m)$ for all $m \in M\backslash \{0\}$

I need to prove equivalent statements of a simple left $R$-module. However, I'm stuck at the following: The map $\phi_m: R/Ann(m) \rightarrow M$, such that $\phi_m(r+Ann(m)) = rm$ for all $r + ...