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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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Intuitive meaning of transitive action

In the context of abstract algebra, following from Dummit's textbook. The action of $G$ on $A$ is called transitive if there is only one orbit. $i.e$, given any two elements $a,b \in A$, there is some $g \in G$ such that $a=g.b$. I want to know…
phy_math
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Explanation for why $1\neq 0$ is explicitly mentioned in Chapter 1 of Spivak's Calculus for properties of numbers.

During the first few pages of Spivak's Calculus (Third edition) in chapter 1 it mentions six properties about numbers. (P1) If $a,b,c$ are any numbers, then $a+(b+c)=(a+b)+c$ (P2) If $a$ is any number then $a+0=0+a=a$ (P3) For every number $a$,…
tcmtan
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Prove that any non-zero-divisor of a finite dimensional algebra has an inverse

Let $A$ be a finite dimensional algebra. Prove that an element of $A$ is invertible iff it is not a zero divisor. Let $a$ be an invertible element, then there exists an element $b$ such that $ab=1$ and assyme that $a$ is a zero divisor, then there…
john1990
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What is the abstract algebraic structure of "Elementary Algebra"?

From what I understand the Set of Complex Numbers is Closed under Addition, Subtraction, Multiplication, Division, Exponentiation, Radicals and Logarithms. I understand this to mean that these binary operations applied to any two elements of this…
jcelios
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Example of algebraic structure that is non distributive for BOTH distributive laws and how to do computation in them?

(Apologies if this one sounds like I have not done much research, or I did not aware already have an answer, but I have been searching everywhere and all of these structures presented here, even including the highly exotic division by zero proposals…
Secret
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Finite, normal extension of odd degree.

Let $\mathbb{Q} \subseteq E$ be a finite normal extension. Prove that if $(E : \mathbb{Q})=n$ is odd then $E\subseteq \mathbb{R}$. My attempt: I am using the fact that a finite normal extension is a splitting field for some polynomial $f$ over…
user140436
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Prove that an algebra is finitely generated iff it is isomorphic to the quotient ring $R[x_1,\dots,x_n]/I$

This is a basic question, but I haven't done abstract algebra in a while and not certain of the answer. We say that an $R$-algebra $A$ is finitely generated if there exist $a_1, \dots, a_n$ such that $A=R[a_1,\dots, a_n]$. I want to prove that an…
user115697
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If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$

I was checking my old set of homework problems that I found this one: If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$. One direction is easy. If $a$ is a cube in $G$ then there exists $y \in G$ such…
user66733
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Can an element other than the neutral element be its own inverse?

Take the following operation $*$ on the set $\{a, b\}$: $a * b = a$ $b * a = a$ $a * a = b$ $b * b = b$ $b$ is the neutral element. Can $a$ also be its own inverse, even though it's not the neutral element? Or does the inverse property require…
MikeRand
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Does a subset $S$ of $\mathbb{Q}$ with the following conditions, equal $\mathbb{Q}$?

Let $S$ be a subset of $\mathbb{Q}$ such that: $0 \in S$; If $x\in S$, then $(x-1) \in S$ and $(x+1) \in S$; If $x \in S \setminus\{0,1\}$, then $\frac{1}{x(x-1)} \in S$. Does $S = \mathbb{Q}$?
Hasan Karimi
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Is it in any way possible to work with reals modulo integer values?

I'm working with a strange and perhaps (mathematically) nonsensical realm. I'd like to know if we can work with $\mathbb{R}/ \mathbb{Z}$. For example, if we take $(\pi \bmod 3)$ we get $(\pi - 3)$. I'm wondering if I always end up with the same…
Matt Groff
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Is there an intuitive proof of the identity $ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x$ from general principles?

Let $R$ be a commutative ring with unity, and let $S\subset R$ be any finite set. Then $$ \sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x,$$ which is easy enough to show by induction. Does this follow from any sort of general…
awwalker
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Why can't there be a quintic formula?

Possible Duplicate: Why is it so hard to find the roots of polynomial equations? For polynomials (with real coefficients), in degrees 2, 3, 4, there are the quadratic, cubic, and quartic formula, though the quartic formula is extremely long, so…
user59677
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Benedict Gross Abstract Algebra

I recently started watching Harvard's Abstract Algebra Course by Benedict Gross. I am not able to find his lecture notes and problem set and they are not given on the Harvard website. Anybody has them or knows where I can find them
user8839370
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Subgroup of finite index contains a normal subgroup of finite index

Let $G$ be a group and $H\leq G$. Suppose $[G:H]$ is finite. Show that there exists a normal subgroup $N \subseteq H$ in $G$ which is also of finite index in $G$. My idea was to use $$N := \bigcap_{g \in G} gHg^{-1}$$ It is clear that $N$ is…
TheGeekGreek
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