Wolfram Alpha:

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Sets, Groups, Modules, Rings and Vector Spaces
----------------------------------------------

Sets
----

A set is a collection of unique elements. On the
other hand, a group is an algebraic structure that
obeys certain axioms (a composition law, closure,
identity, associativity and inverses).

Groups, G
---------

A group, (G,*) is a set of elements gi = {a,b,c ...}
that satisfies the following axioms:

Operations:  + or .

Closure: a + b = c

a.b = c

Commutativity:  a + b = b + a

a.b = b.a  (Abelian)

a.b ≠ b.a  (Non-Abelian)

Associativity:  a + (b + c) = (a + b) + c

a.(b.c) = (a.b).c

Distributivity *:  a.(b + c) = a.b + a.c

* The distributive law requires two operations and
so is not relative to groups.

Additive Identity:  0 + a = a

Multiplicative Identity:  1a = a

Additive Inverse:  a + (-a) = 0

Multiplicative inverse:  a.a-1 = 1

Note:  Every field is a ring but not every ring
is a field.

Subgroups
---------

H is a subgroup of G if it obeys the same axioms
as the group.  Subgroups are denoted by H ≤ G.

H = {e} is the trivial subgrup.

H = G - A group, G, is its own subgroup.

Fields, F
---------

Fields should not be confused with a vector fields.
Here a field is any set of elements, f, that satisfies
the following axioms.

Rigorous definition:

Operations:  + and .

Commutativity:  a + b = b + a

a.b = b.a  (Abelian)

Associativity:  a + (b + c) = (a + b) + c

a.(b.c) = (a.b).c

Distributivity:  a.(b + c) = a.b + a.c

Additive Identity:  0 + a = a

Multiplicative Identity:  1a = a

Additive Inverse:  a + (-a) = 0

Multiplicative inverse:  a.a-1 = 1

Altenative definition:

One can alternatively define a field by 4
and their required properties. Division by
zero is, by definition, excluded.

Example:

Z is not a field because the multiplicave inverse
is ∉ Z, i.e. a = 3 a-1 = 1/3 ∉ Z.

Rings
-----

A ring is a field that is:

1.  Commutative under addition but not necessarily
under multiplication.  If it is, it is called
a commutative ring.

2.  It does not have to have a multiplicative
inverse.

If the elements of a commutative ring have a
multiplicative inverse, the ring is a field, i.e.

Abelian  Abelian  Inverse    Type
-------  -------  -------    ----

Field    Y         Y        Y

Ring     Y         Y        Y       Field
Y         Y        N       Comm. ring
Y         N        N       Ring

Every field is a ring but not every ring is a
field.

Example 1:

-   -         -   -
A = | 0 1 |   B = | 0 1 |
| 1 0 |       | 0 0 |
-   -         -   -

-   -          -   -
AB = | 0 0 |   BA = | 1 0 |
| 0 1 |        | 0 0 |
-   -          -   -

These are Abelian under addition but not under
multiplication.  A is its own inverse and B
has det = 0.  Therefore, these matrices form
a ring.

Example 2:

Z is a ring because the multiplicave inverse is
not a condition.

Vectors Spaces, V
-----------------

A vector space is any set of elements, v, that
satisfy the following axioms..

Operations:  + and scalar multiplication

Commutativity:  v1 + v2 = v2 + v1  (Abelian under +)

Associativity:  v1 + (v2 + v3) = (v1 + v2) + v3

Additive Identity:  0 + v1 = v1

Additive Inverse:  v1 + (-v1) = 0

Field, F, of scalars:

f.v1 = scaled vector

Distributive property:  f.(v1 + v2) = fv1 + fv2

(f1 + f2)v1 = f1v1 + f2v1

Associative property:  f1.(f2.v1) = f1.(f2.v1)

Action of 1:  1v1 = v1

Modules, M
----------

A module is any set of elements, m, that satisfy
the following axioms.

Operations:  + and scalar multiplication

Commutativity:  m1 + m2 = m2 + m1  (Abelian under +)

Associativity:  m1 + (m2 + m3) = (m1 + m2) + m3

Additive Identity:  0 + m1 = m1

Additive Inverse:  m1 + (-m1) = 0

Ring, R, of scalars:

r.m1 = scaled element

Distributive property:  r.(m1 + m2) = rm1 + rm2

(r1 + r2)m1 = r1m1 + r2m1

Associative property:  r1.(r2.m1) = r1.(r2.m1)

Action of 1:  1m1 = m1

Example:

M = R3 = {(x,y,z) | x,y,z ∈ R}

-          -
| a11 a12 a13 |
R = | a21 a22 a23 |
| a31 a32 a33 |
-          -

r ∈ R, m ∈ M

r.m = matrix multiplication

Where the scalar is a linear map, r.

Vector spaces over fields are analagous to modules
over rings.  In the latter case field of rings acts
on the elements of the module.

The general terminology used is a 'vector space over
a field' or a 'module over a ring'.  Modules are very
closely related to the representation theory of groups.```