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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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Fields of fractions

In my textbook there's a theorem that goes like this: Let $D$ be an integral domain. Then $D$ can be embedded in a field of fractions $F_D$, where any element in $F_D$ can be expressed as the quotient of two elements in $D$. Furthermore, the field…
Cay
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The natural map $R\mapsto R[t]/(ft-1)$ is injective

Let $R$ be a commutative ring and $f\in R$ such that $f$ is not a zero divisor (thanks Darij for correction). How can I rigorously show that The canonical map $R\mapsto R[t]/(ft-1)$ is injective. Context: This step shows up in Rabinowitch trick…
Prism
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Finding the size of a permutation group

Let A be an m by n matrix (you can assume that all elements of A are distinct). By a "legitimate transposition" on A, I mean an operation on A that swaps the (i,j)th and (k,l)th elements of A for some i<=k and j<=l. My question is : How many…
Somabha Mukherjee
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Question about Ring Theory (the sum of two ideals is a ring).

A question I recently encountered was: "Let $R$ be a commutative ring with one not equal to zero. Let $I$ and $J$ be ideals of $R$, such that $I + J = R$. Show that for any positive integers a,b, we have that $I^a + J^b = R$. Hint: What is $I\cap…
ThinkConnect
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Prove ${ Z }_{ 3 }{ \times Z }_{ 4 }\simeq { Z }_{ 12 }$ directly.

$${ Z }_{ 3 }{ \times Z }_{ 4 }\simeq { Z }_{ 12 }$$ Above notations are ideal. I tried $f(a,b)=4a+3b$ But, I run into a brick wall because of multiplication for homomorphism. How to prove it?
YSRiemann
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Why do we say the group of order $p^3$ is not always abelian?

For any integer $n$, there is a cyclic group $\Bbb Z(n)$. So for any integer $n$ there is always an abelian group $G$. So why we say the group of order $p^3$ is not always abelian?
avijit kundu
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Proving the set of Orbits is a partition on a set

Let $G$ act on a set $X$. I am trying to prove that the set of orbits is a partition of $X$. I first define a relation $\sim$ on $X$ by $$x \sim y \iff y=x \wedge g$$ for some $g \in G$ Then I show that $\sim$ is a equivalence relation I know the…
Al jabra
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Abelian group and element order

Suppose that $G$ is an Abelian group of order 35 and every element of $G$ satisfies the equation $x^{35} = e.$ Prove that G is cyclic. I know that since very element of $G$ satisfies the equation $x^{35} = e$ so the order of elements are $1,5,7,35$.…
user236626
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counter example for "every ideal is contained in a maximal ideal" in non-unital case?

As known, the fact "every ideal in a unital commutative ring is contained in a maximal ideal" is proven using Zorn's lemma, but it really uses that the ring has the identity. (While using Zorn's lemma, you take a union and to show it's different…
vgmath
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Is $\mathbb{Z}[2\sqrt{2}]$ a PID?

I am practicing for my algebra qual and I would like to know if $\mathbb{Z}[2\sqrt{2}]$ is a PID. I had no intuition at first except the fact that $\mathbb{Z}[i\sqrt{2}]$ is a ED with norm $N(a+i\sqrt{2}b)=a^2+2b^2$. I tried proving that…
Justine
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Prove that every Principal ideal Domain is a Unique Factorization Domain

I know that to prove something is a a Unique factorization Domain i need to show that the factorization is unique. So i start like that.// Proof: Let $P$ be a principal ideal domain, and let $$r \in P$$ where $r \not=0$ and $r$ is not a unit. So…
user146269
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Show that $\mathbb{Z}_4[x]$ has infinitely many units and infinitely many nilpotents.

In $\mathbb{Z}_4$, $1$, $2$, $3$ are all units and nilpotents in additive operation; but only 1, 3 are units and 2 is nilpotent in multiplicative operation. I did some experiments like polynomial combination that I might work out a way to prove…
Shannon
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Let $d$ be a non-square discriminant. If $I \subset \mathcal{O}_d$ is a nonzero ideal, then $I \cong \mathbb{Z}^2$ as an abelian group.

For every integer $d \equiv 0,1 \mod 4,$ we define a quadratic ring $\mathcal{O}_d$ by: If $d \equiv 0 \mod 4,$ let $\mathcal{O}_d = \mathbb{Z}\left[\sqrt{\dfrac{d}{4}} \right].$ If $d$ is a non-square congruent to $1,$ then $\mathcal{O}_d = \left\{…
user167857
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Simple Solving Equations in Groups Problem

From Prof. Charles Pinter's A Book of Abstract Algebra's Chapter 4 exercises: Let $a, b, c$ and $x$ be elements of a group $G$. In each of the following, solve for $x$ in terms of $a, b$ and $c$. Problem 1: $$axb = c$$ I came up with an…
Kevin Meredith
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Number of subfields of finite field

Possible Duplicate: Subfields of finite fields Field with $2^{15}$ elements How many subfields? Including trivial one and the whole thing I guess $15$. Right or not my guess? I think every subfield gives a subspace of dimension dividing $2^{15}$.…
user23086
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