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Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

Abstract algebra is the study of algebraic objects, i.e. sets endowed with one or more operations on the elements of those sets. In particular, the study of abstract algebra considers the algebraic structures and properties of which such operations induce. It can be considered as the generalization of the study of the algebraic structure of the integers and real numbers (arithmetic), or the study of matrices and vector spaces (linear algebra).

Some algebraic objects are monoids, groups, rings, fields, vector spaces, modules, algebras, and categories, among many other less prominent objects.

Examples

  1. The set of non-negative integers $\mathbb{N} = \{0,1,2,3,\dotsc\}$ is a monoid under the operation $+$.

  2. The integers $\mathbb{Z} = \{\dotsc,-1,0,1,\dotsc\}$ under the binary operation of $+$ form a group.

  3. Furthermore, $\mathbb{Z}$ has the structure of a ring when you consider it as being equipped with both addition and multiplication.

  4. The real numbers $\mathbb{R}$ with their usual addition and multiplication form a field.

  5. The set of $n\times n$ matrices with entries in $\mathbb{R}$ with matrix addition and multiplication form a ring.

  6. The set of $1\times n$ vectors over the real numbers, with vector addition, and multiplication by elements of the $n\times n$ real matrices on the right are an example of a module for the ring of matrices.

In addition to studying the objects themselves, abstract algebra considers homomorphisms between the objects and various constructions and tools, which are useful for studying the objects.

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Confusion on order of group action on function

Let $\Gamma$ be a group and $\Gamma'\leq \Gamma$ be the subgroup. Let $Y$ be $\Gamma'$ module. Define induction $Ind^\Gamma_{\Gamma'}Y=$ as abelian group…
user45765
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On finitely generated R-module

Let $M$ be a finitely generated R-module and $I$ be an ideal of $R$ such that $IM=M$. Then prove there exists $a\in I$ such that $(1-a)M=0$. Thanks
maryam
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Galois Group does not depend on the choice of the field extension

Let $K$ be a field, and a polynomial $f\in K[X]$. Let $E$ and $F$ be two splitting fields of $f$ (albeit isomorphic), to show is two Galois groups Gal$(E/K)$ and Gal$(F/K)$ are isomorphic. One idea could be the Galois groups also permute the roots.…
CO2
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Monoid times monoid equal to itself

I am reading about monoids. It says that if $G$ is a commutative monoid, then $GG = G$ because G contains a unit element. If $G = \{1,2,3\}$ and the composition law is multiplication, then $9$ should be in $GG$, but is not in $G$. What am I not…
Jonathan Tooker
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Roots of the cyclotomic polynomial

Let $\Phi_p$ be a cyclotomic polynomial and $q,p$ different primes. I know how many roots $\Phi_p$ has in $\mathbb{Z}/q\mathbb{Z}[X]$ and that these roots are simple. Now I can lift these roots to roots in $\mathbb{Z}_q[X]$ according to Hensel. Is…
james
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What is meant by "Maps" and |.| and "canonical basis" in abstract algebra (in the given context)?

I found this abstract algebra question in a previous test paper: Suppose $\Bbb{F}$ is a field and $\mathrm{X}$ is a non-empty set. Then $\text{Maps}(\mathrm{X},\Bbb{F})$ is a vector space over $\Bbb{F}$. If $|\mathrm{X}|=3$ then find a…
user554252
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Prove $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},A)\cong A[n]$ by using left-exactness of Hom?

Let $A$ be an abelian group, then $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},A)\cong A[n]$, where $A[n]=\{a\in A:na=0\}$. How do I prove this by using left-exactness of Hom? (This is Exercise 2.7 from Rotman's An Introduction to…
Yuxiao Xie
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Suppose $M$ is a module and $K = \{(m, m) : m \in M\} \subset M \oplus M$. Show $K$ is a submodule of $M\oplus M$ which is a direct summand.

I'd like to have some clarification and help on this problem. Suppose $M$ is a module and $K = \{(m, m) : m \in M\} \subset M \oplus M$. Show $K$ is a submodule of $M\oplus M$ which is a direct summand. Showing $K$ to be a submodule in $M \times M$…
randomafk
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Is there mathematical language to discuss groups in terms of their operations, instead of their elements?

For example, in my abstract algebra class, we recently proved that homomorphisms preserved identity; i.e., given groups $(G,*)$ and $(H,\#)$, if $\theta : G \rightarrow H$ is a homomorphism, then $\theta(e_g) = e_h$. This was the…
Eugene
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algebraic conjugates of a sum

Given algebraic numbers $a$ and $b$, is it the case that all algebraic conjugates of $a+b$ take the form $a'+b'$ where $a'$ and $b'$ are algebraic conjugates of $a$ and $b$ respectively?
Alex Kindel
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Which of these rings are integral domains?

Which of the following rings are integral domains? (a) $\{a+b\sqrt{5}:a,b\in \mathbb{Q}\}$ (b) the ring of continuous functions from $[0,1]$ (c) the polynomial ring $\mathbb{Z}[x]$. (d) the ring of complex analytic functions on the disc…
poton
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Proof for the inverse in sub group is as same as inverse in group

I have a question: Let H be the subgroup of G. I want to prove that if for any element that is in the H, the inverse of that element in the group and sub group is the same. I don't know where to start. I know that and know how to prove that…
Nima
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Why is the set $A=\{f\in F:f(0)=0\}$ an ideal where $F=\{f|f:[-1,1]\to \mathbb{R}\}$?

Why is the set $A=\{f\in F:f(0)=0\}$ an ideal where $F=\{f\mid f:[-1,1]\to \mathbb{R}\}$ and $(F,+,\times)$ is the usual commutative ring? I first checked whether $A$ is a subring. For this observe that $A$ is non-empty and for any $f,g\in A$…
Student
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Invertible elements in $k[G]\otimes k[G]$

Let $G$ be a finite group, $k$ a field and $k[G]$ the group algebra. Is something known about the invertible elements in $k[G]\otimes k[G]$? Maybe the isomorphism $k[G]\otimes k[G]\cong k[G\times G]$ is usefull.
user527859
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Relatively prime integer

Please help me to answer the following problem: Let $F$ a field. Show that if $m$ and $n$ are relatively prime integer, and $r,s\in F^{\times}$ satisfy $r^m=s^n$ then there are $u,v\in F^{\times}$ such that $r=u^n$ and $s=v^m$. Thanks
siwar
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