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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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Easy ten points. I can't seem to get what seems to be a very obvious result?

Given module homomorphism $\alpha:M\to N$ and $P\le M$ (submodule). Can we say $\alpha(P)\le \alpha(M)$? In the solution the lecturer points out that the image of $\alpha$ is the same as the image of $\alpha\circ i$ where $i$ is the inclusion map…
Catherine
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Factoring over ring of power series

How would we factor $6+x$ over $\mathbb{Z}[[x]]$, the ring of formal power series with integer coefficients? Proving things are prime is easy, but factoring a nonprime is difficult. Thanks in advance.
Izzy Zheng
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In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5=e$ is a multiple of 4.

I am new to abstract algebra, group theory. So, its difficult for me to solve this question. Kindly, please help me.
user117741
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How can I find the subgroups of $\mathbb{Z}_{20}$?

I have to determine the subgroup lattice for $\mathbb{Z}_{20}$. The order of all subgroups divides the order the group. So, since $|\mathbb{Z}_{20}|=20$, the orders of the subgroups are: $20,10,5,4,2,1$. But how can I find these subgroups??
Mary Star
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What is the significance of closure property on a set and a binary operator?

A group is defined as a set with a binary operation define on it, which satisfies closure, associative, identity and inverse properties. Why is closure property included in the definition what is the significance of this property?
sai prasad
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Find the order of a extension field.

I have the following exercice: Let $K$ be the two element field and $P(X)=X^3+X+1\in K[X]$. Show that $P$ is irreductible in $K[X]$. Let $\alpha$ be a root of $P$ in an extension of $K$. Show that $\alpha^2$ is also a root of $P$. Consider the…
Luc M
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How to determine if a polynomial is irreducible over $\mathbb Z_{3, 7} [x]$ to find its splitting field?

I need to find the splitting field of the polynomial $f(x)=x^4-2$ over $\mathbb Z_3[x]$ and $\mathbb Z_7[x]$. I know that if it's irreducible over $\mathbb Z_3[x]$, then $\mathbb Z_3 [x]/ \langle f(x) \rangle$ is the splitting field. The problem is…
Twnk
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$R$ is a commutative ring with identity. Prove: All of the zero divisors of $R$ including zero, contain at least one prime ideal.

Let's say $A=\{ x \mid x \text{ is a zero divisor}\}\cup\{ 0\}.$Let $a,b\in A$.Let $P$ be an ideal of $A$, i.e. $P\trianglelefteq A$. Now it's necessary to show that: first, $P\ne A$. Second, $ab\in P\Rightarrow a\in P \vee b\in P$. But the question…
Mill
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When is an abelian group (as $\mathbb{Z}$-module) completely reducible (semisimple)?

I know the answer when the abelian group is finitely generated. That is, the direct sum of a series of $\mathbb{Z}_p$, $p$ prime. However, I don't know the case of infinitely generated ablian group. I've seen an example that the direct product of…
Roun
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What exactly is a skew field extension?

I'm reading a text wich says: "Let $k\subset K$ be a skew field extension." Now I know what a field extension is, but I'm not quite sure what to think of a skew field extension. Also, how would you construct such an object?
gebruiker
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$\mathbb{C}/\mathbb{Z} \cong \mathbb{C}-{0}$

I have to prove that $\mathbb{C}/ \mathbb{Z} \cong \mathbb{C}-\{0\}$ holds. I`m using this theorem: If $\phi: \mathbb{C} \rightarrow \mathbb{C}-{0}$ is a homomorphism and $H=Ker(\phi)$, with $H$ as a normal subgroup for $\mathbb{C}$. Then,…
Pim
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Is this assumption "legal" when showing that the circle group is a subgroup of $\mathbb{C}^{\times} $?

As an abstract algebra exercise, I need to show that the $n$th roots of unity is indeed a subgroup of $\mathbb{C}^\times$. The only part of the proof I am concerned about is the multiplicative inverse. Here is what I have stated so far. Let $a$ be…
Joseph DiNatale
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With $G=S_5$ prove that the only proper normal subgroup is $A_5$

I can find proofs (but I admit I couldn't do them without referencing them) that $A_5$ is a normal subgroup of $S_5$, I also know (and the question says I may use - but without is more than welcome) that $A_5$ is simple, that is it's only normal…
Alec Teal
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Simple module over $A$ has finite dimension over $K$

Could someone explain the proof of the following statement to me: Let $A$ be a $K$-algebra. $A$ is finite $\implies$ a simple module over $A$ has finite dimension over $K$ Proof: If $M$ is simple, then $mA\subset M$ is a sub-module, hence we see…
gebruiker
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How does one know this is a subgroup?

I was watching a video of an algebra lecture on line. The substitute teacher was presenting a pf. of a lemma: $$m\mathbb{Z} + n\mathbb{Z} = \mbox{gcd}(m,n)\mathbb{Z}\quad\mbox{where }m,n\in\mathbb{Z}$$ I am a bit puzzled by the development of the…
user12802
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