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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Proposition 5.21 in Atiyah-MacDonald

There's just one step in this proof I can't see for the life of me. Set up: We have a field K and an algebraically closed field $\Omega$. $(B, g)$ is maximal in the set $\Sigma$ of ordered pairs $(A, f)$ where $A$ is a subring of K and $f$ a…
Thank you
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Why does a group homomorphism preserve more structure than a monoid homomorphism while satisfying fewer equations

Is there a deeper (categorical) reason for this? On the one hand a group homomorphism $\phi:(G,\cdot)\to (H,\star)$ preserves 'results of operations' as well as the identity element and inverse elements, but satisfies only one equation: $$\forall…
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Does $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ imply $A\cong B$?

If $A$ and $B$ are abelian groups, do we have that $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ implies $A\cong B$? Motivation: I was just thinking about different ways of deducing equality from expressions by quotienting, then realized I didn't…
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In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct summands of free modules, and these need not be free. On…
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Vector Space Structures over ($\mathbb{R}$,+)

Consider the abelian group ($\mathbb{R}$,+) of real numbers with the usual addition. Is there a scalar multiplication \begin{equation} \cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, \end{equation} other than the usual multiplication,…
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Is it possible to construct a field larger than the complex numbers?

Can we extend the complex numbers in any way such that $\mathbb{C} \subset\mathbb{C}[a]$ ? Or is $\mathbb{C}$ the extension to end all extensions?
AnotherPerson
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If $\gamma\in \Bbb A$ then there exists a $\pm$quadratic coefficiented polynomial for which $\gamma$ is a root.

$\mathbf{1.\space Proposition}$ Let $\gamma$ be a solution to the equation: $$ \sum_{i=0}^n \rm a_i\rm x^i=0, \rm a_i\in\Bbb Z, a_n=1. $$ Then, there exists a polynomial $p\in \Bbb Z[x]$ such that: $$ p=\sum _{i=0}^m\rm s_ix^i $$ Where $\rm…
YoTengoUnLCD
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Approximation Lemma in Serre's Local Fields

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated. Approximation Lemma Let $k$ be a positive integer. For every $i$, $1\leq i \leq k$, let $\mathfrak p_i$ be prime ideals of…
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Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually prove a list of properties about limits of sequences of real and complex numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$ (where $c \in F$ for some field…
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Algebra and Analysis

I'm a math major at university and my tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble understanding the difference between them. What are the actual differences between algebra and…
Lara
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What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But until now, I haven't asked myself about infinite…
rschwieb
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Examples of calculus on "strange" spaces

I am interested in examples of calculus on "strange" spaces. For example, you can take the derivative of a regular expression[1][2]. Also the concept extends past regular languages, to more general formal languages[3]. You can also do calculus on…
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give a counterexample of monoid

If $G$ is a monoid, $e$ is its identity, if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$? If not, please prove $b=c$. Thanks a lot.
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The set of all nilpotent elements is an ideal

Given that R is commutative ring with unity, I want show that set of all nilpotent elements is an ideal of R. I know how to show ideal if set is given but here set is not given to me. Can anyone help me?
Siddhant Trivedi
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Example of a non-Noetherian complete local ring

I was looking for an example of a non-Noetherian complete local commutative ring with $1$. I would appreciate if anyone can point to a reference.
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