Canonical examples of algebraic structures

Question

Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.

"When I read about a [insert structure here], I immediately think of [example]."

Or maybe you think about a small number of examples. For example, when someone says "group", maybe you immediately think of one example of an abelian group and one example of a non-abelian group. Maybe you have a list of examples that you test new theorems on.

I am an analyst by training. When I read algebra, I can follow the logic line by line, but I don't have the repertoire of examples in my head that I have for analysis and so it's hard for me to picture anything.

Answer 1

Neat question...

Abelian Group: Z or Z/nZ

Nonabelian Group: Dihedral groups or GL_n

Commutative Ring: Z or C[x]

Noncommutative Ring: Matrix rings

Division Ring: Hamilton's Quaternions

Field: R or C

Lie Algebra: sl_2

Answer 2

Lots of good answers. I figured I'd throw in a list of non-examples, since these are pretty handy as well. (These are all standard non-examples, nothing fancy.)

A non-Noetherian ring with only one prime ideal: (k[x1, x2, x3, ...]/(xi xj : 1 <= i,j), (x1,x2,...)).

A non-Cohen-Macaulay ring: k[x, y]/(x2, xy).

A category that doesn't have products: the category of fields with field homomorphisms.

A ring which isn't flat over another ring: A = k[x2, x3] and B = k[x].

Two non-zero rings whose tensor product is zero: Z2 and Z3

Answer 3

I often think of "universal examples". This is useful because then you can actually prove something in the general case - at least theoretically - just by looking at these examples.

Semigroup: $\mathbb{N}$ with $+$ or $*$

Group: Automorphism groups of sets ($Sym(n)$) or of polyhedra (e.g. $D(n)$).

Virtual cyclic group: Semidirect products $\mathbb{Z} \rtimes \mathbb{Z}/n$.

Abelian group: $\mathbb{Z}^n$

Non-finitely generated group: $\mathbb{Q}$

Divisible group: $\mathbb{Q}/\mathbb{Z}$

Ring: $\mathbb{Z}[x_1,...,x_n]$

Graded ring: Singular cohomology of a space.

Ring without unit: $2\mathbb{Z}$, $C_0(\mathbb{N})$

Non-commutative ring: Endomorphisms of abelian groups, such as $M_n(\mathbb{Z})$.

Non-noetherian ring: $\mathbb{Z}[x_1,x_2,...]$.

Ring with zero divisors: $\mathbb{Z}[x]/x^2$

Principal ideal domain which is not euclidean: $\mathbb{Z}[(1+\sqrt{-19})/2]$

Finite ring: $\mathbb{F}_2^n$.

Local ring: Fields, and the $p$-adics $\mathbb{Z}_p$

Non-smooth $k$-algebra: $k[x,y]/(x^2-y^3)$

Field: $\mathbb{Q}, \mathbb{F}_p$

Field extension: $\mathbb{Q}(i) / \mathbb{Q}, k(t)/k$

Module: sections of a vector bundle. Free <=> trivial. Point <=> vector space.

Flat / non-flat module: $\mathbb{Q}$ and $\mathbb{Z}/2$ over $\mathbb{Z}$

Locally free, but not free module: $(2,1+\sqrt{-5})$ over $\mathbb{Z}[\sqrt{-5}]$

... perhaps I should stop here, this is an infinite list.

Answer 4

A more abstract version of "matrix ring" is the endomorphism ring of a module. If you take your module to be a free module over R, then you get matrix rings, but there are plenty of other examples of modules that are worth thinking about. This is my go-to example when I need a ring that isn't necessarily commutative.

There was another question asking for various examples of modules. Besides the free modules, the next-easiest R-modules for a ring R are ideals I and quotient rings R/I. In particular, remembering that abelian groups are Z-modules is useful.

The standard "geometric" principal ideal domain is k[X], for k a field. The standard "geometric" UFD is a polynomial ring over a field (or over a UFD). So if you want a UFD that isn't a PID, you have a bunch of choices, like k[X, Y] or Z[X]. If you want an integral domain that isn't a UFD, you can think of the coordinate ring of a generic affine variety.

Answer 5

Projective module: given a vector bundle over a compact hausdorff space, the vector space of continuous sections of the bundle is a projective module over the algebra of continuous functions on the space.

Answer 6

I'll mention some more useful non-examples:

A non-monoid semigroup: $({\mathbb Z^+},+)$.

A non-group monoid: $({\mathbb N}, +)$, $({\mathbb Z},\cdot)$.

A non-integral domain: ${\mathbb Z}_6$.

A noetherian non-artinian ring: the integers ${\mathbb Z}$, the ring of Laurent polynomials over a field $K[x,x^{-1}]$.

A non-unital semisimple ring: the row-finite, column-finite, infinite matrices over a field ${\mathbb M}_{\infty}(K)$.

A simple non-semisimple algebra: The Weyl algebra $K[X,Y]/(XY-YX-1)$.

A non-simple indecomposable ring: $K[x,x^{-1}]$.

Answer 7

Not an algebraist myself, but this is interesting. Few things I do know:

Read ------------------------------- Quickly think of

algebraically closed field --------- $\mathbb C$

abelian group ---------------------- $(\mathbb R;+)$

non-abelian group ----------------- $(\operatorname{SO}(n,\mathbb R);\times)$ ; ($n>2$)

simple group ----------------------- $(A_n; \circ)$ ; ($n\ge5$)

Answer 8

More from the Lie-algebraic realm:

simple Lie algebra: of course, the ubiquitous sl(2) 
solvable Lie algebra: < x,y | [x,y]=x >                   
nilpotent Lie algebra: < x,y,z | [x,y]=z, [x,z]=[y,z]=0 >
simple Lie algebra in characteristic p: < e_i, i\in Z/pZ | [e_i,e_j]=(j-i)e_{i+j} > (Witt algebra)
semisimple Lie algebra in characteristic p: sl(2)\otimes K[t]/(t^p) + Kd/dt
Kac-Moody algebra: sl(2)\otimes C[t,t^{-1}] + Ctd/dt + Cz 

associative commutative algebra: K[t]/(t^n)

algebraic group: SL_2(K)

an algebraic object given by generators and relations: the corresponding 2-generated free object 
(e.g., 2-generated free group, 2-generated free Lie algebra, etc.)

Answer 9

Euclidean domain: Z[i] (Gaussian integers) Principal ideal domain: ring of integers in Q(\sqrt{-19}) Unique factorization domains: Z[x], C[x,y] Finite field: F_4=F_2[t]/(t^2+t+1)

Answer 10

The set of non-zero octonions is the canonical Moufang loop.

Answer 11

This is, at least, how it went for me just now:

Read:                Think:

field                Q(sqrt 2), or some other number field
finite group         symmetric group
Lie group            the torus R^2/Z^2
ring                 a matrix ring over... some other ring
commutative ring     a polynomial ring k[x_1,...,x_n]
Lie algebra          sl(2)

Answer 12

This is a fun one! Here are some of my favorite examples (I'll try not to list ones that have already been listed).

Finite Group : SL_n(Z/pZ).

Nonabelian group (simple) : Free groups, Baumslag-Solitar groups

Nonabelian group (hard) : SL_n(Z), Aut(F_n), mapping class group

Lie algebra : Depending on context, either sl_n(C) or a free Lie algebra

Algebraic Variety : twisted cubic, a hyperelliptic curve

Answer 13

$p$-group: the Sylow $p$-subgroups of $S_n$ or $GL_n(\mathbb{F}_p)$.

The former can be described as follows: split $n$ up into $\lfloor \frac{n}{p} \rfloor$ blocks of size $p$ and a remainder, and allow the permutations that only permute individual blocks. Then split the blocks themselves into "2-blocks" of size $p$ and allow permutations that permute the 2-blocks etc. The result is an iterated wreath product.

The latter can be described as the subgroup of upper-triangular matrices with all ones on the diagonal, e.g. a Heisenberg group.

Answer 14

Pasha mentions presentations of groups; in this connection, I think there are few better examples than the Coxeter groups, with such beauties as $\langle a, b : a^2 = b^2 = 1, (ab)^3 = 1\rangle \cong S_3$.

In keeping with the theme of non-examples, I remember being puzzled by the notions of reductivity and quasi-split-ness until someone told me that a Borel subgroup of an algebraic group (like the group of invertible, upper-triangular $n \times n$ matrices) isn't the former, and the multiplicative group of a skew field (considered as an algebraic group over its centre) isn't the latter.

Answer 15

I said this somewhere else before. But when people talk about projective limit, inverse limit, I think about $\mathbb{Z}_{p}$ as the projective limit of $\mathbb{Z}/p^{n}\mathbb{Z}$. When people talk about modular forms, which example do you usually think? Why?

Answer 16

The question title is about "Canonical examples of algebraic structures", but the text of the question (and most answers) make(s) me think I can probably answer in more generality to: "What do you do in front of a new (set of) definition(s)?", or "What do you take when a theorem asks to consider ...?".

I generally first check with too trivial examples, before I go to trivial but probably more interesting examples (as found in other answers) ; and in a third pass, I even consider objects which don't have one of the wanted properties (counter-examples, also discussed in other anwers). So let me tell more about that first step, since it was overlooked so far.

Of course, the defect is that for some situations you end up with an example so simple that it doesn't give anything. But those same correct but too primitive examples can later turn up as interesting and enlightening counter-examples in other definitions/results (I give two such situations below the list), and that gives them some merit.

Here is a short list of witless examples, to demonstrate how weak those first examples can/should be:

• A set: $\emptyset$ ;
• A monoid: {0} ;
• A group: {0} ;
• A ring: {0} ;
• A vector space: {0} ;
• An algebra: {0} ;
• An element in a structure : the zero or unit, if it fits the bill (if it doesn't, its time will come in the third step)
• A morphism: the zero map, if it fits the bill (if it doesn't... see above) or the identity ;
• A symmetry in an object: the identity (or -id) ;
• A projection in an object: the identity or the zero map ;
• A graded something: either grade with the zero ring or $\mathbb{Z}/2\mathbb{Z}$ ;
• A ring of integers in a number field: $\mathbb{Z}$ ;
• (not 100% algebraic) A distance on a space: the one which is zero on a diagonal pair and one outside ;
• (the same idea, but in a more algebraic setting) A valuation on a field: the trivial one ;

Here is a first example of the usefulness of being thickheaded: the definition of integral ring excludes the zero ring, and the definition of a prime element in a ring excludes the units. Both are linked to the question of the {0} ring, through the definition of a prime ideal by giving an integral quotient.

Now for a second such situation: finding an example of distance which doesn't come from a norm is trivial if your very first idea of a distance is the foolish one ; the same question is hard if you take honest examples from the second step, where you'll probably only have the usual suspects on $\mathbb{R}^2$.

Answer 17

I think you'll find out very different my example of algebraic structures.

Monoid: the monoid of words over a finite alphabet.

Group: the group of words over a finite alphabet.

Ring: the polinomial ring.

Module: $\mathbb K^n$ for some $\mathbb K$ field.

Category: the path category over some graph $\mathbb G$.

$R$-Algebra (for some ring $R$): $M_n(R)$, i.e. the ring of matrix over the ring $R$.

These is exactly those algebraic structures that come in my mind when I think/try to prove fact about algebraic structures. Why? Because they're free object and are the terms models of their corresponding algebraic theories, so every other algebraic models derive from these by adding relations.