Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
1 answer

Ring with specific property is a division ring

I'm struggling with the following exercise in my algebra course: Let $A$ be a non-trivial ring not necessarily with multiplicative identity. Suppose that for every $a \in A \setminus \left\{ {0}\right\}$ there exists an unique $a' \in A$ such…
u1571372
  • 3,205
3
votes
0 answers

Calculating the commutator subgroup.

I want to find the commutator subgroup ($G^k$, $k=1,2,..$) of such matrix group:$$ \left(\begin{array}{ccc} 1 & * & * \\ 0 & * & 0 \\ 0 & * & * \\ \end{array}\right)$$ I have multiplied them "by hands" and I got that $G^1= …
xxxxx
  • 869
3
votes
3 answers

Primes in $\mathbb Z$ and $\mathbb Z[i]$

Give an example with justification of two primes $p$ and $q$ in $\mathbb Z$ such that $p$ is a prime in $\mathbb Z[i]$ but $q$ is not a prime in $\mathbb Z[i]$. I know that $\mathbb Z[i]$ is the set of Gaussian integers which also form a ring…
Sourav Chakraborty
  • 371
3
votes
2 answers

A ring which is equal to its quotient ring

Let the ring $S \cong \mathbb Z \times \mathbb Z$ and $I= \{(x,0): x\text{ is in }\mathbb Z\}$ then every $(x,y)$ in $\mathbb Z \times \mathbb Z$ can be written as $(0,y) + (x,0)$ which is an element of $(\mathbb Z\times \mathbb Z)/I$. Then $S\cong…
Sourav Chakraborty
  • 371
3
votes
1 answer

Comaximal ideals

I am confused with relatively prime, comaximal ideals (sum of the two ideals is the full ring) in polynomial ring in $n$ variables. Are the following statements true? 1) Let $I=(f(x,y,z))$ and $J=(g(x,y,z))$ be two principal ideals (in a polynomial…
MathStudent
  • 1,816
3
votes
1 answer

Algebraic and Transcendental Elements

Let $F$ ba an extension extension of the field $K$. An element $a$ in $F$ is said to be algebraic if $a$ is the root of some non-zero polynomial $f \in K[x]$. Using only the definition, how do we prove that for a field of rational functions,…
Guest_000
  • 809
3
votes
2 answers

The field of rationals inside an algebraically closed field

Let $F$ be an algebraically closed field of characteristic zero. Is there a unique copy of the field of rational numbers $\mathbb{Q}$ inside $F$ or there are many? PS: By a copy of $\mathbb{Q}$ I mean a subfield of $F$ which looks like (i.e.…
user48900
  • 1,248
3
votes
1 answer

Prime Ideal in a Ring

I need help with the following question: Let $R$ be a commutative ring, $S \subset R \ \ $ closed under multiplication. define $\Omega $ as the set of all ideal $I$ in $R$ such that $I \cap S = \phi$ proof that if P is maximal under containing in…
Shai Alkoby
  • 133
3
votes
1 answer

Proving a polynomial is algebraic over a tower of fields

Let $K\subseteq E\subseteq F$ be fields. Prove that if $F$ is algebraic over $K$ then $F$ is algebraic over $E$ and $E$ is algebraic over $K$. Proof: Suppose that $F$ is algebraic over $K$. We want to show that $F$ is algebraic over $E$. It follows…
user60887
  • 2,935
3
votes
1 answer

A cycle which is the product of disjoint $m$-cycles does not necessarily have order $m$?

Suppose a cycle $\sigma\in S_n$ has decomposition a product of disjoint $p$-cycles, for $p$ a prime. So each cycle has order $p$, and thus $\sigma^p=id$, so $\sigma$ has order dividing $p$, and thus order $p$ when $\sigma\neq id$. What if $\sigma$…
KBomberman
  • 33
3
votes
1 answer

In the ring $A[x]$ the Jacobson radical is equal to the nilradical.

Let $A$ be a commutative ring with identity and $N(A[x]):=$ the nilradical of $A[x]$, $J(A[x]):=$ the Jacobson radical of $A[x]$. Then we prove $N(A[x])=J(A[x])$. First, it's obvious that $N(A[x])\subset{J(A[x])}$. So just need to prove…
gaoxinge
  • 4,434
3
votes
2 answers

Does $g = x^m - 1 \mid x^{mk} - 1$ for any $k \in \mathbb{N}$?

Let $g = x^m - 1 \in \mathbb{Z}[x]$. Question: Does $g = x^m - 1 \mid x^{mk} - 1$ for any $k \in \mathbb{N}$? I know of only how to show the much weaker claim that $x^m - 1 \mid x^{m \cdot 2^k} - 1$ for any $k \in \mathbb{N}$. Proof of Weaker…
user115803
  • 125
3
votes
1 answer

How many rotations are there in $\mathbb R^3$ which take $C$ to itself?

Let $C$ denotes the cube $[-1,1]^3\subset\mathbb R^3.$ How many rotations are there in $\mathbb R^3$ which take $C$ to itself? A. $6$ B. $12$ C. $18$ D. $24$
user113578
  • 536
3
votes
2 answers

Prove $A=\{\,ra+na\mid r\in R, n\in\Bbb{Z}\,\}$ is an ideal.

Let $R$ be a commutative ring, and let $a\in R$. Prove that $A=\{\,ra+na\mid r\in R, n\in\Bbb{Z}\,\}$ is an ideal belonging to $R$. Remember that we cannot assume that the ring is unital. I got stumped on this question, and could not proceed. My…
crazydude
  • 31
3
votes
0 answers

Which group is meant by $\mathbb Z/2 \times \mathbb Z/2.$

There is a question in my question paper (Q. No. 11 - Part B) which tells to verify whether The automorphism group $\text{Aut} (\mathbb Z/2 \times \mathbb Z/2)$ is abelian. I don't understand which group is meant here by $\mathbb Z/2 \times…
user113578
  • 536
1 2 3
…
99 100