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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3
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1answer
25 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 ...
2
votes
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 $\mathbb{Z}$, ...
1
vote
1answer
26 views

Why $R/AB$ is cyclic?

Why $R/AB$ is cyclic when $A,B$ are ideals of $R$? I know the a cyclic module is a module is that can be written as $Rm$ and $Rm$ is isomorphic to $R/Ann(m)$. However, I can't see why the module ...
0
votes
0answers
23 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) = ...
7
votes
0answers
27 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 $IJ = I \cap J$, ...
1
vote
3answers
75 views

Show that $R/(I \cap J) \cong (R/I) \times (R/J) $

My question actually follows from this one: Show that if $I + J = R$, then $R/(I \cap J) \cong R/I \times R/J$ What I don't understand is why is it necessary for $I+J=R$, in order for $$ ...
16
votes
5answers
3k views

Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
7
votes
0answers
56 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 ...
-1
votes
0answers
23 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 ...
0
votes
0answers
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 ...
0
votes
1answer
86 views

Is this element algebraic over $\mathbb{Q}$?

I'm trying to see whether the following element is algebraic over $\mathbb{Q}$ and if it is - try and find its degree: $$ a = \left(\frac{(1 + \sqrt[3]{7})^{7/5}}{(\sqrt[3]{7} - 7)^3 + ...
3
votes
1answer
19 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.
2
votes
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 ...
1
vote
1answer
21 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 ...
5
votes
1answer
63 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 ...
4
votes
3answers
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 ...
0
votes
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. ...
2
votes
2answers
155 views

Converse to Chinese Remainder Theorem

So as seen on this question Converse of the Chinese Remainder Theorem, we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, ...
-2
votes
0answers
34 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 $15$?
2
votes
2answers
106 views

Can $R \times R$ be isomorphic to $R$ as rings?

I know from this question that $R \times R$ can be isomorphic to $R$, as $R$-modules. But can they ever be isomorphic as rings?
2
votes
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 ...
0
votes
0answers
9 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 ...
0
votes
0answers
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 ...
0
votes
0answers
26 views

Quotient of Ideals in matrix rings

I'd like to know where could I find some info about the quotient $I:J=\{a\in R\mid aJ\subseteq I\}$ ($R$ a ring) in matrix rings? Or for example, in a matrix ring over $\mathbb{Z}$. I would like to ...
1
vote
0answers
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 ...
0
votes
0answers
11 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 $GL$?
0
votes
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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votes
0answers
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.
3
votes
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 ...
0
votes
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 ...
6
votes
2answers
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 ...
2
votes
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, ...
0
votes
0answers
15 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 ...
2
votes
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 ...
1
vote
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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votes
0answers
26 views

Is the algebra of these circuits valid?

I drew these circuits when I was studying Boolean Algebra. Is the algebra of these circuits valid?
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2answers
138 views
+250

Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains

For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that ...
2
votes
1answer
44 views

Subgroups of finite index have finitely many conjugates

is it true the following statement: Let $G$ be a group and let $H$ be a subgroup of $G$. If the index $[G:H]$ of $H$ in $G$ is finite, then $H$ have finitely many conjugates. What I think, is that ...
0
votes
3answers
2k views

How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? [duplicate]

How do I find the number of group homomorphisms from the symmetric group $S_3$ to $\mathbb{Z}/6\mathbb{Z}$?
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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 $\phi$: $G$ $\mapsto$ $G/H$ ...
2
votes
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 ...
3
votes
2answers
2k views

Example of prime, not maximal ideal

This is a homework from videolecture: Show that $(x^2-y)$ is prime but not maximal in $\mathbb C[x,y]$". Linked SE pages offer to approach this by proving that $\mathbb C[x,y]/(x^2-y)$ is an ...
1
vote
1answer
28 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 ...
0
votes
0answers
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 ...
6
votes
2answers
659 views

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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0answers
10 views
1
vote
1answer
57 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 ...
6
votes
1answer
626 views

Projective and injective modules; direct sums and products

I need two counterexamples. First, a direct sum of $R$-modules is projective iff each one is projective. But I need an example to show that, “an arbitrary direct product of projective modules need ...
1
vote
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 ...
4
votes
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 ...