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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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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8 views

One-dimensional Noetherian UFD is a PID

Hi I am looking for a reference which has a self-contained (elementary; that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does ...
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11 views

If $A$ is an submodule of $B$ and $B$ is an submodule of $C$, is $A$ an submodule of $C$?

Let $C$ be a commutative ring (with 1, if this matters). If $A$ is an submodule of $B$ and $B$ is an submodule of $C$, is $A$ an submodule of $C$? I can't really prove that it is true because it is ...
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1answer
20 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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1answer
39 views

$[U \cap H : G \cap H] \le [U :G]$

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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25 views

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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24 views

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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29 views

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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8 views

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
34 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 ...
3
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1answer
21 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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1answer
22 views

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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2answers
28 views

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
49 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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1answer
22 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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36 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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12 views

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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7 views

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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38 views

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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14 views

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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26 views

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
39 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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1answer
45 views

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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1answer
16 views

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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1answer
13 views

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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16 views

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 cannot find the Smith Normal Form of : $M = \begin{pmatrix} 21 & 0 & 1 \\ 8& 4 & 1\\ 3& 8 ...
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77 views

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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57 views

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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1answer
17 views

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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1answer
33 views

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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1answer
45 views

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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1answer
64 views

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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0answers
23 views

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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13 views

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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1answer
60 views

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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1answer
33 views

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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1answer
36 views

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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55 views

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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1answer
29 views

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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18 views

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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1answer
24 views

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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1answer
21 views

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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21 views

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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42 views

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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1answer
25 views

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 + ...
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30 views

Find all non-trivial submodules of a direct sum of two non-isomorphic simple modules

Let $R$ be a ring with $1$. Let $M_1$ and $M_2$ be two non-isomorphic simple (nonzero) $R$-modules. Find all non-trivial submodules of $M_1 \bigoplus M_2$. Solution: $M_1 \bigoplus M_2 \cong M_1 ...
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33 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
3
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2answers
41 views

If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$

A problem from my algebra text: If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$ I think it's false since $x = 0 + 0i = 0 \in \mathbb{Z}[i]$ is not a unit, but $0 + 0 ...
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1answer
34 views

Let $R$ be a non-commutative ring. Show that if $R$ is simple and has 1, then $Z(R) = \{a \in R | ra = ar$ for all $r \in R \}$ is a field.

Let $R$ be a non-commutative ring. Show that if $R$ is simple and has 1, then $Z(R) = \{a \in R | ra = ar$ for all $r \in R \}$ is a field. I think what I need to do is to show that $Z(R)$ is simple ...