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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 $1-\zeta_n$ a unit in every ring where $n$ is invertible?

Let $R$ be a commutative ring where $n\ge 2$ is invertible and containing a primitive $n$th root of 1, called $\zeta_n$, satisfying $\zeta_n^n = 1$ and $\zeta_n^k\ne 1$ for any $1\le k\le n$. Is $1-\zeta_n$ invertible on $R$? Thanks to Hurkyl's…
user355183
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What is a set of generators for the multiplicative group of rationals?

I'm trying to think of a set of generators for the group $(\mathbb{Q^\times, *})$, which I think is countably generated (as opposed to finitely generated). (I wonder if such a set could be the primes of $\mathbb{Z}$ (together with $1$ and $-1),$…
étale-cohomology
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Non-abelian groups that are "closest" to being abelian

Given a finite, non-abelian group $G$ of order $n$, how close can $G$ be to being abelian? More formally, define the following density measure: $$d(G) = \frac{\#\{(a, b) \mid a, b \in G, ab = ba\}}{n^2}$$ The question is, then, are there known upper…
MT_
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Problem with Ring $\mathbb{Z}_p[i]$ and integral domains

Let $$\Bbb Z_p[i]:=\{a+bi\;:\; a,b \in \Bbb Z_p\,\,,\,\, i^2 = -1\}$$ -(a)Show that if $p$ is not prime, then $\mathbb{Z}_p[i]$ is not an integral domain. -(b)Assume $p$ is prime. Show that every nonzero element in $\mathbb{Z}_p[i]$ is a unit if and…
user40105
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How to show simply that a field is a vector space over any of its subfields?

I know that any field is a vector space over itself, see e.g. here -- Prove that the field F is a vector space over itself. Is there a correspondingly simple argument showing that this is the case for any subfield? For example, $\mathbb{C}$ can be…
Chill2Macht
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Proof of first isomorphism theorem of group

(Proof of (first) Isomorphism Theorem) Let $f : G \rightarrow H$ be a surjective group homomorphism. Let $K = \operatorname{ker} f$. Then the map $f' : G/K \rightarrow H$ by $f'(gK) = f(g)$ is well-defined and is an isomorphism. Proof: If $g'K…
Absolutio Zaka
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How to prove that a mapping is homomorphic

Let $ f:(A, \cdot) \to (B, \ast) $ and $g:(B,\ast) \to (C,\times)$ be Operation preserving maps. Then I must prove that $ g \circ f$ is an operation preserving map too. This is what I have so far: Since $f$ is a homomorphism $(A, \cdot)$ and $(B,…
math101
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Multiplicative identity being equal to additive identity in a field

Is it even possible? What consequences would this have if it is possible? My attempt: Let us call this hypothetical universal identity $e$. Fields require distributivity, right? $(a-a)a^{-1} = aa^{-1} - aa^{-1} = e-e = e$ But calculating without…
mathreadler
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Between complex numbers and quaternions?

Complex numbers are $a+ b i $; Quaternions are $a + b i + c j + d k $. So, do there exist numbers like $a + b i + c j$? Here $a$, $b$, $c$, $d$ are all real. Did Hamilton consider such a case?
kaiser
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Find every ring homomorphism between $\mathbb Z$ and $\mathbb Q$

One of the problems I got is nothing but this sentence. No hints, no context. In one direction, a function $\varphi:\mathbb Z\to\mathbb Q$ is a morphism iff $$\forall\star\in\{+,\times\}\quad \forall a,b\in\mathbb Z\quad\varphi(a\star_{\mathbb…
Luke
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Cohesive picture of groups, rings, fields, modules and vector spaces.

If I understand my algebra correctly every field is a ring and every ring is a group, so when we define modules over rings and vector spaces over fields, we then have that every vector space is a module? A linear algebra is defined in Hoffman's book…
Samuel Reid
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If $ab=ba$, Prove $a^2$ commutes with $b^2$

From Dr. Pinter's "A Book of Abstract Algebra": Given $a$ and $b$ are in $G$ and $ab=ba$, we say that $a$ and $b$ commute. Prove $a^2$ commutes with $b^2$ I tried: $$ ab=ba $$ $$ a^{2}b^{2}=b^{2}a^{2} \text{ // to prove that $a^2$ commutes with…
Kevin Meredith
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If $H$ is a cyclic group of even order, $H$ has exactly two elements which square to $1.$

If $H$ is a cyclic group of even order, then $H$ has exactly two elements which square to $1.$ This was used in a answer (Pete Clark's answer) here: Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions but I am not sure…
St Vincent
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No isomorphism between $k[x,y]/(x^{2}-y^{5})$ and ring of polynomials in one indeterminate

Let $k$ be an algebraically closed field. Why there is no isomorphism (as finitely generated $k$-algebras) between the ring $k[x,y]/(x^{2}-y^{5})$ and $k[t]$?
user6495
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Cyclic group Zp

How to show if $p$ is prime, then group $Z_{p}^{*}$ is cyclic. Tip Let $g$ and $h$ of a commutative group $G$ have orders $n$ and $m$ respectively. There exists and element $x \in G$ of order $LCM (n,m)$ In any field $\mathbb{K}$ a polynomial $f…
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