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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 there a name for this identity-like object?

I was grading for a linear algebra class just now and someone remarked that since $U + U = U$, then $U$ would be an identity for the set of subspaces of some vector space with binary operation subspace addition. Obviously I corrected their mistake,…
Christian
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Lie bracket of two derivations. Is it composition of derivations or is it multiplication inside the algebra

Let $A$ be a $K$-algebra ($K$ a field). Let $\delta:A\to A$ be a $K$-derivation. We say that these $K$-derivations form a Lie algebra by the commutator bracket. But $[\delta_1,\delta_2]=\delta_1\delta_2-\delta_2\delta_1$, is this by composition or…
user462339
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Problem 1.5.11 in Aluffi's Algebra: Chapter 0

I am trying to prove the following: Let $A$ and $B$ be sets endowed with the equivalence relations $\sim_A$ and $\sim_B$, respectfully. Define the relation $\sim$ on $A \times B$ by setting $$(a_1,b_1) \sim (a_2,b_2) \; \text{if and only if} \;…
Holdsworth88
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Show that $\Bbb Q/\Bbb Z $ is an infinite group?

How do I show that $\Bbb Q/\Bbb Z $ is an infinite group? I've been thinking that all elements in $\Bbb Q/\Bbb Z $ can be written as $\left[\frac ab\right]$, where a $\in \Bbb Z$ and b $\in \Bbb N \setminus\{0\}$, is this right? Then I've been…
MBrown
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localized module tensor with localized ring

Let $f: S^{-1} M \to S^{-1}A \otimes_A M$ defined by $$m/s \to 1/s \otimes m$$ $g: S^{-1}A \otimes_A M \to S^{-1} M$ defined by $$a/s \otimes m \to am/s $$ Prove that $f$ and $g$ are well defined ? How can we prove $f$ is an…
user48931
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What is structure?

The definitions of structure that I've read define it "a set containing mathematical objects endowed with an operation that enriches its structure." The inclusion of the word being defined in the definition is confusing. I've read several articles…
Hal
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Every Finite Group is Isomorphic to a Subgroup of $A_n$

How does one prove that every finite group is isomorphic to a subgroup of an alternating group?
user109871
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Herstein Problem No.13 Page 109

Give an example of a finite non-abelian group $G$ which contains a subgroup $H_0 \neq (e)$ such that $H_0 \subset H$ for all subgroups $H \neq (e)$ of $G$. Can someone help me please?
user444830
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If a field contains odd nth roots of unity, it contains 2nth roots of unity

Clearly, if $\zeta_{n}$ is a root of unity for $n$ odd, then $\zeta_{n}^{2}$ is also a primitive $n$th root, since $(n, 2) = 1$. Hence, $\zeta_{n}$ is a $2n$th root of unity. But how do I know all the $2n$th roots are in this field, including the…
BMac
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Show that S9 doesn't contain an element of order 18

I first note that 18 = 1 x 18 = 2 x 9 = 3 x 6. Hence an element of order 18 is either an 18-cycles permutations,or a product of a 2-cycles and 9-cycles or a product of a 3-cycles and 6-cycles . Since the first two cases are impossible, and the…
Quy Xuan Cao
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If M irreducible R-module, then M is isomorphic to R/I for I a maximal ideal of R

If M irreducible R-module, then M is isomorphic to R/I for I a maximal ideal of R. [If M is irreducible there is a natural map $R \rightarrow M$ defined by $r \mapsto rm$, where m is any fixed nonzero element of M.] I am self-studying Abstract…
metalder9
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How many different groups whose order is 16 that every element to the fourth power is unit element?

Consider a group $G$ whose order is 16, and every element $x$ satisfies $x^4=e$, where $e$ is the unit element. I am asked how many different groups are there under isomorphism. I only learned the structure of finite abelian groups. And 16-group is…
Edward Wang
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If $f(x, \beta) = 0$ then $f(x, y)$ is divisible by $(y - \beta)$.

I am trying to prove the following statement: Let $k$ be an arbitrary field. Let $f(x, y) \in k[x, y]$ and $\beta \in k$. If $f(x, \beta) = 0 \in k[x]$, then $f(x, y)$ is divisible by $(y - \beta)$. Here is my attempt: Consider $f$ as an element…
nowhere
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How many elements of order $4$ are there in $\mathbb{Z}/2$ × $\mathbb{Z}/4$ × $\mathbb{ Z}/6$?

So my thinking is that obviously $\mathbb{Z}/2$ and $\mathbb{Z}/6$ don't have elements of order $4$ (Lagrange's theorem) but I can look at the order of $2$ for both these groups?
Gragbow
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Determine the minimum degree of f such that is guaranteed to have at least one irrational root

May you tell me if my answer is correct? Thank you so much! Here is the problem: Given f(x)= where the constant term is the product of r distinct primes, determine the minimum degree of f(x) such that is guaranteed to have at least one irrational…
Beginner
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