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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.

85022 questions
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Finite Rings have Invariant Dimension Property

I need help this question Prove that every finite ring has the Invariant Dimension Property (IDP). (Assume $1_R \neq 0_R$.) This is what I know I should do. Let $X$ and $Y$ be two sets such that the free module with basis $X$ is isomorphic to the…
Josh
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Show that the ring $R$ of entire functions does not form a Unique Factorization Domain

Show that the ring $R$ of entire functions does not form a Unique Factorization Domain (U.F.D) My try: I will first check whether $R$ forms an Integral Domain then check whether it is Factorization Domain and ultimately a U.F.D. I.D. Let $f,g\in R$…
Learnmore
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Example of a non-commutative rings with identity that do not contain non-trival ideals and are not division rings

I'm looking for an example of a non-commutative ring, $R$, with identity s.t $R$ does not contain a non-trival 2 sided ideal and $R$ is not a division ring
Mykie
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Irreducible polynomial is always separable in char 0 field

I am reading separable extension. I don't quite understand why $\text{char} 0$ fields must be perfect. If $F$ is a $\text{char} 0$ field, $f(x)$ is an irreducible polynomial in $F[x]$. Then why $f$ must be separable? Assume $f$ has multiple roots,…
cali
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Prove that the following is a Field

Let $p$ be a prime Let $n$ be an element in $Z_{p}^*$ where $n\not=\pm1$. Define a ring structure on $F = Z_p \times Z_p$. We define the addition by $$ (a_1,b_1) + (a_2,b_2) = (a_1 + a_2, b_1 + b_2). $$ And define multiplication by: $$ (a_1,b_1)…
user121615
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Prove that the set A satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity and the negative.

A set $A$ with operation of addition and multiplication is given. Prove that the set $A$ satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity and the negative of an arbitrary $a$. $A$ is the set…
Chilanie
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Polynomial Combinations in $F[x]$

Supposed $f(x), g(x) \in F[x]$ for some field $F$ are polynomials of degrees $m, n $ respectively. Moreover assume that they are relatively prime. By Euclidean algorithm I can find $a'(x), b'(x)$ such that $$a'(x)f(x) + b'(x)g(x) = 1$$ and thus for…
Alex
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Show that $a + b\sqrt{n}$ is an irreducible element of $\mathbb{Z}[\sqrt{n}]$

Let $n(\neq 0,1)$ be a square-free integer. Suppose that $|a^2 - nb^2|$ is a prime integer for $a,b \in \mathbb{Z}$. Show that $a + b\sqrt{n}$ is an irreducible element of $\mathbb{Z}[\sqrt{n}]$. Then, show that $p = |a^2-nb^2|$ is not irreducible…
user10024395
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In $\Bbb Z[\sqrt{2}]=\{a+b\sqrt{2}\rvert a,b∈\Bbb Z\}$, show that every element of the form $(3+2\sqrt{2})^n$ is a unit, where n is a positive integer

In $\Bbb Z[\sqrt{2}]=\{a+b\sqrt{2}\rvert a,b∈\Bbb Z\}$, show that every element of the form $(3+2\sqrt{2})^n$ is a unit, where n is a positive integer. My understanding of a unit is that if a is a unit, then ab = 1 = ba for some b. In other…
frierfly
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If $f(g) = g^k$ is a homomorphism on a finite group $G$ and $k < |G|$ does not divide $|G|$, must $G$ be abelian?

There was a question that asked to prove that if $f(g) = g^3$ is a homomorphism on a finite group $G$ and $3$ does not divide $|G|$, then $G$ is abelian. Does this extend to any $k < |G|$ where $k$ does not divide $|G|$, instead of just the case $k…
user2566092
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Prove the function $f:\Bbb C^{*} \rightarrow \Bbb R^+$ given by $f(a+bi)=\sqrt{a^2+b^2}$ is a homomorphism and describe its kernel.

Prove the function $f:\Bbb C^{*} \rightarrow \Bbb R^+$ given by $f(a+bi)=\sqrt{a^2+b^2}$ is a homomorphism and describe its kernel. Homomorphism: Let $a,b,c,d$ be in $\Bbb C^{*}$ and for $f$ to be a homomorphism then…
Chilanie
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General Concept of an Isomorphism

Suppose that two algebraic structures $(A,+, \cdot, ...)$ and $(B,+, \cdot, ...)$ are isomorphic (... in this case referring to any amount of n-ary operations on the sets A and B). Besides the fact that we have a way to relate the products of any…
AnotherPerson
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Question about Eisenstein Criterion

The following is a use of eisenstein criterion that i have taken out from my lecture note. $f(x, y) = x^4 +x^3y^2 +x^2y^3 +y$ is irreducible in Q[x, y]. This can be proved by treating Q[x,y] as (Q[y])[x] and applying the Eisenstein criterion with p…
user10024395
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Let H be a subgroup of a group G. H is normal iff aH=Ha for every a $\in$ G

Let $H$ be a subgroup of a group $G$. $H$ is normal iff $aH=Ha$ for every $a \in G$ I'm having trouble with this proof both ways actually. Assume $H$ is a subgroup of a group $G$ and $H$ is normal. Since $H$ is normal then for $h \in H$ there exists…
Mark
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example of infinite torsion abelian group

I am new to the concept of torsion. Is there any example for an infinite torsion abelian group? Here is my example: rotation with a rational degree in a clock. Is this an example? Thank you very much!
breezeintopl
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