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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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Is $\mathbb Z[i]/n\mathbb Z[i]$ an integral domain?

Let $ \mathbb{Z}[i]$ denote the ring of the Gaussian intergers. For which of the following value of n is the quotient ring $ \mathbb{Z}[i]/n\mathbb{Z}[i]$ an integral domain? $ a. 2$ $ b. 13$ $ c. 19$ $ d. 7$ I'm doubtful with the following attempt…
Sriti Mallick
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Factor Rings of Polynomial Rings.

Let $\mathbb F$ be a field and $\mathbb F[x]$ the ring of polynomials with coefficients in $\mathbb F$. Let $p(x)$ be an irreducible polynomial in $\mathbb F[x]$. Let $k$ be a positive integer and consider the vector space $V$, over the field…
zacarias
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Some Results in $\mathbb{Z} [\sqrt{10}]$

This is a question from an old Oxford undergrad paper on calculations in $\mathbb{Z} [\sqrt{10}]$. We equip this ring with the Eucliden function $d(a+b\sqrt{10})=|a^2-10b^2|$. I want to prove the following results: If $d(x)=1$, then $\frac{1}{x}…
Mathmo
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What is $\mbox{Hom}(\mathbb{Z}/2,\mathbb{Z}/n)$?

Simple question (I seem have asked a few like this...) What is $\mbox{Hom}(\mathbb{Z}/2,\mathbb{Z}/n)$? (for $n \ne 2$)
Juan S
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Find all finite groups $G$ s.t for any $a,b\in G$ either $a$ is a power of $b$ or $b$ is a power of $a$

Find all finite groups $G$ s.t for any $a,b\in G$ either $a$ is a power of $b$ or $b$ is a power of $a$ I think i showed that all such groups are $Z_{p^n}$ for $p$ prime, is this correct? I first showed that the group must be cyclic by considering…
2132123
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Since $K/F$ is Galois, every embedding of $K$ fixing $F$ is an automorphism of $K$?

I'm trying to read a proof in Dummit and Foote of the statement Suppose $K/F$ is a Galois extension and $F'/F$ is any extension. Then $KF'/F'$ is a Galois extension, and $Gal(KF'/F') \cong Gal(K/K \cap F')$. One line I am confused about is Since…
badatmath
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If $\phi(\alpha)$ is prime in $\mathbb{Z}$, show that $\alpha$ is prime in $\mathbb{Z}[i]$

If $\alpha=a+bi$ is a Gaussian integer, let $\phi(\alpha)=a^2+b^2$. If $\phi(\alpha)$ is prime in $\mathbb{Z}$, show that $\alpha$ is prime in $\mathbb{Z}[i]$. I use the idea that if $a^2+b^2=p$ where $p$ is prime number, then…
Idonknow
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Nicolas Boubarki, Algebra I, Chapter 1, § 2, Ex. 12

Nicolas Boubarki, Algebra I, Chapter 1, § 2, Ex. 12: ($E$ is a Semigroup with associative law (represented multiplicatively), $\gamma_a(x)=ax$.) Under a multiplicative law on $E$, let $ a \in E $ be such that $\gamma_a $ is surjective. (a) Show…
nature1729
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common left and right coset representatives for a subgroup of finite index

Assume that $G$ is a group, and that $H$ is a (not necessarily normal) subgroup of $G$ having finite index $r=[G:H]$. A subset $\{x_1,\cdots,x_r\}\subset{G}$ is called a left transversal of $H$ in $G$ provided that $\{x_1{H},\cdots,x_r{H}\}$ is…
student
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Equational theory of sedenions and beyond

Consider a Cayley-Dickson algebra $(X,+,*,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, from quaternions to octonions, we lose…
user107952
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What is conjugate?

The concept of conjugate seems to exist in many fields of mathermatic such as complex conjugate, group conjugate, etc. I search through many websites about what exactly is the conjugate. Most of them always claim that "just change the sign". Suppose…
Kanokpon Arm
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Abstract algebra polynomial problem

Let $F$ be any field and $a,b\in F,\,\,a\neq b$. Find the greatest common divisor of $f(x) = x + a$ and $g(x) = x + b$. Since the degree of both is $1$, the gcd is $1$ or $f(x)$ or $g(x)$, since $a\neq b$. So $\gcd(f(x),g(x))=1$. Am I right for…
user67584
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Field contained in a division ring lies in the center of the ring

Given a division ring $R$. A field $F$ is contained in $R$. Show that $F \subseteq Z(R)$. I think this statement is correct, but I find it difficult for me to prove it. I am aware of the fact that for a division ring $R$, the center of $R$, $Z(R)$…
user693282
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Not a Zero Divisor

Let $R$ be a commutative ring. Then we say $a \in R$ is a zero divisor if there exists $b \neq 0$ such that $ab = 0$. I want to know what it means to not be a zero divisor. So I tried to negate the statement: $a$ is not a zero divisor if for every…
Student
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Prove that this ring is an integral domain based on newly defined binary operations

"Define a new addition and multiplication on $\mathbb Z$ by the rules: $a(+) b = a + b – 1$ and $a(*) b = ab – (a + b) + 2$. Prove that with these new binary operations $\mathbb Z$ is an integral domain. You may assume that under these new…
mike
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