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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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Ideals in a Quotient Ring

Let $R$ be a commutative ring with identity and $I$ na ideal of $R$. It is well known that there the is a one-to-one correspondence between the set of all ideals of $R$ which contain $I$ and the set of all ideals of $R/I$. In this correspondence to…
zacarias
  • 3,158
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Difference between function and polynomials

I would like to know if there is an important difference between functions and polynomials? Because in essence they seem very similar, because of the evaluation homomorphism. Thanks in advance.
user42912
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Trying to prove structure result for ${\rm Hom}(A,B)$

Let $A$ be a $\mathbb Z$-algebra that is finitely-generated and free as a $\mathbb Z$-module and let $\pi: A \rightarrow \mathbb Z$ a nontrivial $\mathbb Z$-module homomorphism. For a positive integer, $m$, I am interested in ${\rm Hom}_{\,\mathbb…
Duncan
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Why does $\mathbb C [x,y]= \mathbb C[x+iy,x-iy]$?

In a proof that I am studying, the author makes a substitution/ change of variables. and claims $\mathbb C [x,y]= \mathbb C[x+iy,x-iy]$. But how can one rigorously show this? I have a few problems with it. First, is it really the case that these…
violet
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Same arithmetic

This question come to my mind with this example: Let's say $a\oplus b = \min(a,b)$ and $a\otimes b = a+b$. We will work in $S = \mathbb{R} \cup \{\infty \}$. They do tropical geometry with this. But some sources do same thing with this definitions:…
user87853
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I don't understand the proof of the second isomorphism theorem.

I don't understand how the author gets the last couple of lines. I understand, by the homomorphism theorem(perhaps sometimes called "The First Ismoprhism Theorem"), $\bar{G} \cong G/N$, but how do they then conclude $(G/N)/(K/N) \cong G/K$? Is the…
user1923
  • 41
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Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain

Fraleigh Section 23 12.Give an example to show that a factor ring of an integral domain may be a field 13.Give an example to show that a factor ring of an integral domain may have divisors of $0$ 14.Give an example to show that a factor ring of a…
InfimumMaximum
  • 1,406
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Modules over $\mathbb Q[x]$ that are $2$-dimensional as vector spaces over $\mathbb Q$

I would like to find all $\mathbb Q[x]$-modules that are $2$-dimensional as vector spaces over $\mathbb Q$. I do not even know where to start. Answers or any suggestions would help. Thanks.
stella
  • 149
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For which of $m \in \{5, 6\}$ is $\mathbb Z/m\mathbb Z$ a field?

I have the following problem that I need to solve: For which of $m \in \{5, 6\}$ is $\mathbb Z/m\mathbb Z$ a field? I know how to do this for $m = 5$. I know that $[1][1] = [1]$, $[2][3] = [1]$, $[4][4] = [1]$. So every non-zero element of…
Amber
  • 175
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constraints of using orbit stabilizer theorem

Theorem: For a group $G$ and set $X$, let $\alpha:G\times X\to X$ be an action, where $\alpha(g,x):= g \cdot x$. $\forall x\in X$, $f:G/G_{x}\to G\cdot x$, $f(gG_{x})=g\cdot x$ is a well defined bijection. Is this theorem true for all group…
Irene
  • 519
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Clarification need for the meaning of "underlying set" in the context of concrete categories in Hungerford's graduate Algebra text.

For the following example taken from Hungerford graduate Algebra text, can someone provide clarification where the example says: "However, in the third example after Definition 7.1, is the function $\sigma$ assigns to the group $G$ that usual…
Seth
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$R[x]/(x^n-1)=R[G]$ as rings

Let $R$ be a commmutative ring with $1$ and $G$ finite cyclic group of order $n$. Show that $R[x]/(x^n-1)=R[G]$ (isomorphic) as rings. This is what I did. Suppose $G=\langle b\rangle $. Let $\psi\colon R[x] \to R[G]$ by…
bluerose
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The units of $\mathbb Z_4[x]$

The units of $\mathbb Z_4[x]$ should be $$\{f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n\mid a_0\in\{1,3\},a_i\in\{0,2\}, 1\le i\le n\}.$$ But I don't know how to prove. Any suggestion?
Yeyeye
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How are symmetric functions related to the proof of the Abel-Ruffini theorem?

Good afternoon. I don't speak English. Therefore, there may be errors in the text. I found a proof of the Abel-Ruffini theorem, without using the Galois theory. This is the original proof, translated into modern mathematical language with minor…
alxmdf
  • 31
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In which abstract structure is difference of left and right inverse obvious.

Apart from the title, have below question too. As left and right inverse coincide for bijective functions, so in groups there should be always a single inverse. It should be only for proving if the given abstract structure is a group or not, need…
jiten
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