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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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What is after field?

If, a group is a set, $G$, together with an operation addition, where an operation is a mapping that associates an element of the set to every pair of its elements, satisfying some requirements known as the group axioms. If, a field is a set $F$…
athos
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If $(x_n)$ is a linear recurrence, is the same true for the subsequence $(x_{pn+q})$?

Let $R$ be a ring and let $M$ be a left $R$-module. Then a linear recurrence in $M$ is a sequence $(x_n)_{n\geq 0}$ in $M$ for which there exist scalars $r_0,\ldots,r_d\in R$ such that $$x_{n+d+1} = r_0x_0 + \cdots r_{n+d}x_{n+d}$$ for all $n\geq…
Ehsaan
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Commutative ring $R$ such that $R[x]$ has nonconstant units.

I have an exercise asking to give an example of a commutative ring $R$ such that $R[x]$ has nonconstant units. At first glance, surely $\mathbb{Z}[x]$ is a commutative ring and would give nonconstant units, since the multiplicative inverse of a…
kaiserphellos
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Is the set of positive real numbers a ring under these operations?

Let $L$ be the set of positive real numbers. Two operations are defined: $$ a \oplus b = ab$$ $$a \times b = a^{\log b}.$$ Is L a ring? 1) a $\oplus$ b = ab, ab $\in$ L. 2) a x b = $ a^{\log b}, a^{\log b} \in L$. 3) addition is commutative 4)1 is…
grayQuant
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If all the roots of a polynomial in $\mathbb{Q}[x]$ are integers, then polynomial is in $\mathbb{Z}[x]$

Prove that if all the roots of a polynomial in $\mathbb{Q}[x]$ are integers, then polynomial is in $\mathbb{Z}[x]$ Efforts: Let $p(x)=a_0+a_1x +\dots a_nx^n$ be a polynomial in $Q[x]$ We are given that $p(x)$ has all roots in $Z$ so…
Shweta Aggrawal
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Is there a field structure on $\mathbb{R}^3$ that keeps its structure as a vector space over $\mathbb{Q}$？

This was a problem from my abstract algebra final. My professor wanted us to show that there is NO multiplication on $\mathbb{R}^3$ appropriately defined that makes it a field together with the natural addition in the form of vectors. I've seen a…
NEne
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Prove the following is a subring

Let $R=\{m+n\sqrt{2} \mid m,n\in \mathbb{Z}\}$. Prove that $R$ is a subring of the real numbers. I just want to know how to get started really. My professor has used the same example for the past two months on everything we've done and hasn't…
USC
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finite odd order abelian group property

question: let $|G|=odd $ where $G$ is a finite commutative group then to show every element of $G$ is a square. ans 1> to show that $∀g∈G,∃g_1∈G,g=g_1^2$. let $g \in G$ then $|g|$ $\big |$ $|G| \implies |g|=2n+1$ for some $n$…
jim
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On the definition of ideal (is it a subring?)

My book defines an ideal to be a subring of R such that $xr \in I$ and $rx \in I$, whenever $r \in R$ and $x \in I$. However, by definition any subring of a ring with unity must contain unity. So it follows that in any ring with unity, the only…
user61924
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Question about ring homomorphism.

I have a ring homomorphism $f : \mathbb{Z}[x] \to \mathbb{Z}$ such that $f$ sends $x$ to $3$. I want to find $f(x^3+2)$. My attempt is, since $f$ is a homomorphism $$f(x^3+2)=f^3(x)+f(2)=27+f(2)$$ but what is $f(2)$?
paradox
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If $u$ is a unit in $R$. Then is it necessary that it will also be a unit in $R[x]$?

If $u$ is a unit of $R$, then it means that an inverse belongs to $R$. Thus it belongs to any ring containing $R$. Is there a better answer to this question?
LM10
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Confusion about the statement that the preimage of a prime ideal under ring homomorphism is prime.

The proof that the preimage of a prime ideal $P \subset S$ under say the ring homomorphism $\phi: R \rightarrow S$ is prime is straightforward. However, it only seems to show that if $xy \in \phi^{-1}(P)$ then $x$ or $y \in \phi^{-1}(P).$ Shouldn't…
green frog
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What do we refer to when we say algebraic structure?

A set of all the operations on set $A$ is called an algebraic structure on set $A$? Can anybody explain this statement to me. What I understand from this is that, consider a set $\mathbb{N} $ then, Algebraic structure on set $\mathbb{N}$ is the…
William
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On the maximal subgroup

Let $G$ be a group, $M$ be a maximal subgroup of $G$ and $\alpha \in \operatorname{Aut}(G)$. I want to show that $\alpha(M)$ is a maximal subgroup of $G$. I know $\alpha(M)= \lbrace \alpha(m) \mid m \in M \rbrace$. Suppose contrary that $\alpha(M)…
Hana
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Should all polynomial functions from F to F (F is a field) have coefficients in F?

I'm studying abstract algebra for the first time. let $F$ be a field. I learned that polynomials over field $F$ means polynomials with coefficients in $F$ (denoted by $F[x]$). But when I keep studying, I get to confused the word, 'polynomial…
byster
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