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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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Determining cardinality and units of $\Bbb Z_3[i]$

Let $\Bbb Z_3[i] = \{a + bi \mid a, b ∈ \Bbb Z_3, i^2 = −1\}$ and denote $·_3, +_3, −_3$ the multiplication, the addition and the subtraction $mod$ $3$. With these notations, the addition and the multiplication in $\Bbb Z_3[i]$ are defined as…
TfwBear
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Showing the following coordinate rings are not isomorphic

How do I show that $\frac{k[x,y,z]}{\langle xz-y^2\rangle}$ is not isomorphic to $k[x,y,z]$? (I want to understand the coordinate ring of this surface) Is there a 'natural' evaluation map that realizes $\langle xz-y^2\rangle$ as its kernel?
Herband
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Meaning of $\mathbb{Z}\left[\sqrt{3}\right]$?

I've found here (first answer) this expression: $\mathbb{Z}\left[\sqrt{3}\right]$, where $\mathbb{Z}$ stands for the set of integers. What is the meaning of the expression?
Lo Scrondo
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Let $H,K\leq G$ and $(H,K)$ be the subgroup generated by $\{hkh^{-1}k^{-1}\,|\,h\in H,k\in K\}$. Show that $(H,K)\triangleleft H\vee K$

I working on this Exercise from Algebra by Hungerford (Exercise II.5.3(a)). It states If $H$ and $K$ are subgroups of a group $G$, let $(H,K)$ be the subgroup of $G$ generated by the elements $\{hkh^{-1}k^{-1}\,|\,h\in H,k\in K\}$. Show that…
Blake
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Metric on ring of formal power series

Consider the ring of formal power series $R[[x]]$ and given $\sum a_{n}x^{n}$ define a metric on $R[[x]]$ as follows: $d((a_n),(b_n))=2^{-k}$ where $k$ is the smallest natural number such that $a_{k} \neq b_{k}$ (if no such $k$ exists we define…
user31509
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A Fraction field problem

Let $R$ be an integral domain (in particular commutative ring) with identity and let $X$ be a sub field of $R^{frac}$ (fraction field of $R$). Is it true that there exists a sub ring $A$ of $R$ such that $X=A^{frac}$?
users31526
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Linearly disjoint?

Let $f$ and $g$ be irreducible polynomials over a field $K$ with $\deg f=\deg g =3$ and let the discriminant of $f$ be positive and the discriminant of $g$ negative. Does it follow that the splitting fields of $f$ and $g$ are linearly disjoint? If…
Lisa Mainhard
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Product of $\geq 2^r$ elements can be written as a simple product of $r$ elements?

This comes from Jacobson's Basic Algebra I, Exercise 4 of Section 1.4, found on page 42. I don't understand the following problem. For a given binary composition define a simple product of the sequence of elements $a_1,a_2,\dots,a_n$ inductively as…
Adelaide Dokras
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Condition on semiring to ensure that $f: k \rightarrow k+k$ is injective

Let $(S,+,\cdot)$ be a semiring with or without 0 but necessarily with 1. Let $f: S \rightarrow S$ be defined by $f(k)=k+k$. What is the weakest possible extra assumption I need to make on $S$ so that $f$ is injective. "Weak" will mean an…
Jackson Walters
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how to give a group structure

suppose you have two sets $G_1$ and $G_2$ with same cardinality in $G_2$ you have the group structure and there is a bijective map from $G_1$ to $G_2$ this is just a set map. can we define a binary operation on $G_1$ with the help of binary…
Myshkin
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Additive inverse of multiplicative identity, must it be it's own multiplicative inverse?

Ok another probably very basic algebra question. With real numbers and integers and complex numbers, one is used to $(-1) \cdot (-1) = 1$, i.e. the additive inverse of the multiplicative identity is it's own multiplicative inverse. Does this have to…
mathreadler
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A question about class equation

If $G$ is a group, then class equation is given by $$|G|=|Z(G)|+ \sum_{i=1}^K[G:C_{G}(x_i)] $$ where $x_i\notin Z(G)$ For dihedral group $D_8$, class equation is given by $$2+2+2+2$$ but there is another way to write this is $$1+1+2+2+2$$ My…
user278346
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Are there any typically studied structures (say $\mathfrak D$) where the additive identity, $0_\mathfrak D$ has a multiplicative inverse?

Are there any typically studied structures (say $\mathfrak D$) where the additive identity, $0_\mathfrak D$ has a multiplicative inverse? In most usual settings, i.e, complex and real numbers as fields, ring of matrices, etc, this isn't true, what…
YoTengoUnLCD
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Why is such an ideal ambiguous?

Suppose I have an $R$-ideal $I$ with $$I=(1-\zeta)^n XR$$ with $R=\mathbb{Z}[\zeta]$, $\zeta$ a primitive $p$-th root, $X$ an ideal in the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\zeta + \zeta^{-1}]$ (would it be $\mathbb{Z}[\zeta +…
Mike
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Intuition surrounding units in $R[x]$

My lecture notes state that an 'easy' result is If $R$ is an integral domain then an irreducible element of $R$ remains irreducible in $R[x]$, and the units in $R$ and in $R[x]$ are the same. I can't seem to get my head around why this is the…
user26069
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