Contents
Overview
Three levels of abstraction in algebra are
 concrete algebra: specific elements of a specific set with precisely
defined operations.
 abstract algebra / axiomatic algebra: general algebraic structures with
operations satisfying postulates, instead of specific sets.
 universal algebra: "relief" from specifying postulates.
Algebraic Structures
The following algebraic structures together with their corresponding
homomorphisms form categories. Also all algebraic structures share the following
terminology:
Let (E,⋅) have any algebraic structure T.
 subT
 U≤E :⇔ U⊆E ∧ (U,⋅_{U×U}) has itself an algebraic
structure T.
U is called a subT then.
 isomorphism

Let E,F have the algebraic structure T,
and φ:E→F be a map. 

If 

Then φ is a 
iso 
∃ψ:F→E homomorphism with φ∘ψ=id_{F}
∧ ψ∘φ=id_{E} 

isomorphism of T
⊂ homomorphism of T 
This condition for isomorphisms will often be equivalent to φ being a
bijective homomorphism.
Note
Just like the definition above, several of the
following definitions for algebraic structures and homomorphisms are meant
cumulative. This is written symbolically as f.ex. "group ⊂ monoid"
which means that a set is a group if it is a monoid and satisfies the additional
properties of groups (which would be (i) in this particular case). In turn, a
set is a monoid if it is a semigroup and satisfies (n), etc. This notation saves
a lot of space and is convenient for usual cases.
Groups
Algebraic structures with one law of composition
Let M be a set. 

If 

Then M is a 
C 
s 
⋅:M×M→M is a law of composition 
stable 
magma 
Mag 
a 
∀x,y,z∈M (x⋅y)⋅z = x⋅(y⋅z) 
associative 
semigroup ⊂ magma 
Sgp 
n 
∃1∈M ∀x∈M 1⋅x = x = x⋅1 
neutral 
monoid ⊂ semigroup 
Mon 
i 
∀x∈M ∃x^{1}∈M x⋅x^{1}
= 1 = x^{1}⋅x 
inverses 
group ⊂ monoid 
Gr 
c 
∀x,y∈M x⋅y = y⋅x 
commutative 
Abelian group ⊂ group 
Ab 
Note
There are names for other structures as well, like unital magma for a magma
satisfying (n), but they are less common. In case of multiplication the sign ⋅
is often omitted. In fact, it is always true that a set can have at most one
neutral element, and at most one (left and right) inverse of any element.
M^{×} := {x∈M ¦ x has an inverse} is called the unit group
of the monoid M.
Homomorphisms of algebraic structures with one law of composition
Let (M,⋅), (N,∗) be
magmas/semigroups/monoids/groups/Abelian groups, and φ:M→N a map. 

If 

Then φ is a 
m 
∀x,y∈M φ(x⋅y) = φ(x)∗φ(y) 
morph 
homomorphism of ... 
1 
φ(1) = 1 
unital 
homomorphism of monoids ⊂
homomorphism of semigroups 
Note
Remember that the definitions for homomorphisms
are meant cumulative. This means that for φ to be a homomorphism of monoids,
both (m) and (1) must be true. In fact, although both of these must also be true
for homomorphisms of both groups, the property (1) already follows from (m)
provided that M and N are groups. So the property (1) only must be verified for
monoids, which is the justification for the suggestive notation in the table
above.
If φ:M→N is a homomorphism (of magmas), it carries much of the algebraic
structure of M over to φ(M)⊆N. The image (φ(M),∗) inherits the properties
(s),(a),(n),(i),(c),(d) from (M,⋅). However in
general if N is not a group and φ not unital it may be φ(1)≠1 (and thus φ(x^{1})≠φ(x)^{1}).
If φ:M→N is a homomorphism of T, then Im(φ):=φ(M)≤N
and Ker(φ):=^{1}φ({0})⊴M are subT
(and Ker(φ) even is a normal subgroup resp. an ideal for groups resp. rings or
algebras). Also for N'≤N, Ker(φ)⊆^{1}φ(N')≤M is a subgroup/ring
(and
even a normal subgroup resp. ideal if N' is).
{H≤G/N}→̃{N⊆H≤G} is bijective (for subgroups/normal subgroups/subrings/ideals).
Examples
 For every set X there is the free magma M(X) is the free
algebra of terms over {⋅} generated by X which equals the set of binary
trees with leafs marked with elements of X and composition of two trees
under a new root.
 For every set X there is the free semigroup (of words=lists) F^{a}(X)
:= X^{+} := ⋃_{n∈N\{0}} X^{n}.
 For every set X there is the free monoid (of words) Mo(X) := X^{*}
:= ⋃_{n∈N} X^{n}.
 For every set X there is the free commutative monoid N^{(X)}.
 For every set X there is the free group F(X) with basis X where
∀y∈F(X) ∃!x_{1}^{ε1}⋅...⋅x_{n}^{εn}
= y with x_{i}∈X ∧ ε_{i}∈{1,1} ∧ x_{i}^{εi}≠x_{i+1}^{εi+1}.
 Ab≅ZMod.
 The Cartesian product ∏_{i∈I} A_{i} := {(a_{i})_{i∈I}
¦ a_{i}∈A_{i}} with componentwise laws (requiring AxCh)
is the product in Ens,Mag,Sgp,Mon,Gr,Ab,Rng,Rng1,cRng,RMod,KVec,RAlg,RAlgas,RAlg1,...
(but Fld has no product).
Especially A^{I} := ∏_{i∈I} A.
 The direct sum ⊕_{i∈I} A_{i} := {(a_{i})_{i∈I}
_{}∈ ∏_{i∈I} A_{i} ¦ a_{i}=1 p.t. i∈I}
with componentwise laws is the coproduct in cMon,
Mon??,Ab,RMod,KVec,...,
but that is not the coproduct in Gr
nor in RAlg.
Especially A^{(I)} := ⊕_{i∈I} A.
 (X^{X},∘) is a monoid for any set X, and S_{X} := Perm(X)
:= (X^{X})^{×} ≤ X is the symmetric group. If M is a
magma, then Aut(M)≤S_{M}.
 If M is a magma and C a commutative T,
then Hom(M,C)≤C^{M} is a commutative subT.
(T is one of Sgp,Mon,Gr,Ab,RMod,KVec,
R is a commutative? ring ).
Groups and actions
Algebraic structures with a law of composition and a law of action
(also see below)
Let Ω,E be sets. 

If 

Then E is a 
C 
s 
 ·:Ω→E^{E} is an action of Ω on E.
 ·:Ω×E→E is a law of left action with operators Ω.
 ·:E×Ω→E is a law of right action with operators Ω.

action 
"set with action" 


 (E,⋅) is a group
 ^:Ω→E^{E} is an action of Ω on E.
 s ^ is distributive over ⋅, i.e.
 ∀α∈Ω ∀x,y∈E (x⋅y)^{α} = (x^{α})
⋅ (y^{α}).
 ∀α∈Ω ^^{α} is an endomorphism of E.


group with operators ⊂
group∩"set with action" 
GpOpΩ 
Note
Groups are identified with groups with operators
in ∅. By an abuse of language, laws of left actions are simply called laws of
actions.
Homomorphisms of algebraic structures with one law of composition and
a law of action
Let M,N be groups with operators in Ω,
and φ:M→N a map. 

If 

Then φ is a 
m 
∀α∈Ω ∀x∈M φ(x^{α}) = φ(x)^{α} 
morph 
homomorphism of groups with operators ⊂
homomorphism of groups 
Groups operating on a set
Let G be a group, E a set. 

If 

Then E is a 
C 

 (G,⋅) is a group
 ·:G→E^{E} is a homomorphism of monoids (left
operation), i.e.
 ∀α,β∈G ∀x∈E α·(β·x) = (α⋅β)·x.
 ∀x∈E 1·x = x.


left Gset 
GEns 

 (G,⋅) is a group
 ·:G→E^{E} is a homomorphism of monoids into the opposite
monoid of E^{E} (right operation), i.e.
 ∀α,β∈G ∀x∈E (x·α)·β = x·(α⋅β).
 ∀x∈E x·1 = x.


right Gset 

Note
G is called transformation group of E. Also
compare with homomorphisms of groups to see that for groups
operating on a set, the action even is a map ·:G→S_{E}.
Homomorphisms of groups operating on a set
Let E,E' be Gsets, and φ:E→E' a map. 

If 

Then φ is a 
m 
∀α∈G ∀x∈E φ(α·x) = α·φ(x) 
morph 
homomorphism of Gsets 
Examples
 _·x is a homomorphism of Gsets (or Gmorphism) of G (operating on itself
by left translation α·x:=α⋅x). G·x = Im(_·x)⊆E is the orbit
of x∈E in E and
 G\E := E/(y∈G·x) = {G·x ¦ x∈E} quotient of E for left actions.
 E/G := E/(y∈x·G) = {x·G ¦ x∈E} for right actions.
G_{x} := {g∈G ¦ g·x=x}≤G is the fixgroup
of x∈E under G (in this case of group operations equaling the stabilizer).
Orbits and fixgroups are compatible with quotients: ∀x∈x̅∈G\E G·x=G·x̅,G_{x}=G_{x̅}.
Fixgroups and orbits are related by G·x = [G:G_{x}].
Rings
Algebraic structures with two laws of composition
Let R be a set. 

If 
Then R is a 
C 

 (R,+) is an Abelian group
 (R,⋅) is a semigroup
 (d) ⋅ is distributive over +, i.e.
 ∀x,y,z∈R x⋅(y+z) = (x⋅y) + (x⋅z)
 ∀x,y,z∈R (x+y)⋅z = (x⋅z) + (y⋅z)

pseudoring 

n 
(R,⋅) is a monoid 
ring with 1 ⊂ pseudoring 
Rng1 
c 
(R,⋅) is commutative 
commutative ring with 1 ⊂ ring with 1 
cRng1 
i 
(R\{0},⋅) is a group, resp. R^{×} = R\{0} 
skew field ⊂ ring with 1 


(R\{0},⋅) is a commutative group. 
field
= commutative ring with 1
∩ skew field 
Fld 
Note
For a ring R, the neutral element in (R,+) is written as 0, the neutral
element in (R,⋅) as 1.
☡Rings with 1 are simply
called rings, here. However, be careful about the distinction of pseudorings
and true rings with 1. Some authors also refer to a ring when they think of what
we called a pseudoring, and will always speak of a ring with 1 if they talk
about true rings with 1.
A pseudoring R has the following simple properties
 ∀x∈R x⋅0 = 0 = 0⋅x
 ∀x,y∈R x⋅(y) = (x⋅y) = (x)⋅y
 ∀x,y∈R (x)⋅(y) = x⋅y
 R≠{0} ⇔ 1≠0
Homomorphisms of algebraic structures with two laws of composition
Let (R,+,⋅), (R',+,⋅) be
pseudorings/rings with 1/commutative rings/skew fields/fields, and φ:R→R'
a map. 

If 

Then φ is a 
m 
 φ:(R,+)→(R',+) is a homomorphism of groups.
 φ:(R,⋅)→(R',⋅) is a homomorphism of semigroups.

morph 
homomorphism of pseudorings 
1 
φ(1) = 1, i.e. φ:(R,⋅)→(R',⋅) is a homomorphism
of monoids. 
unital 
homomorphism of ... ⊂
homomorphism of pseudoring 
Note
Rng1→Grp;
(R,+,⋅)↦R^{×} is a covariant functor implying that all
homomorphisms φ:R→R' of rings with 1 satisfy φ(R^{×})⊆(R')^{×}.
Therefore, homomorphisms of rings with 1 from a skewfield R are injective. Also
compare with homomorphisms of groups to see that if R
and R' are skewfields, then for homomorphisms of rings the property (1) follows
from (m).
Examples
 ZAlgas≅PseudoRing,
ZAlgasu≅Rng1,
ZAlgascu≅cRng1.
 The tensor product R⊗_{Z}R' is a coproduct in the
category of commutative rings with 1. But no coproduct exists in the
category of rings???
Modules
Algebraic structures with a law of composition and a law of action
Let R be a (pseudo)ring, M a set. 

If 
Then M is a 
C 

 (M,+) is an Abelian group
 ·:R×M→M is a law of action with operators R.
 d · is distributive over +, i.e.
 ∀α∈R ∀x,y∈M
α·(x+y)
= (α·x) + (α·y)
 ∴ modules are commutative groups with operators.
 d ∀α,β∈R ∀x∈M
(α+β)·x = (α·x)
+ (β·x)
 a ∀α,β∈A ∀x∈M
α·(β·x) = (α⋅β)·x
(for left Rpseudomodules)

left Rpseudomodule 


The same as for left Rmodules, except that
 a ∀α,β∈A ∀x∈M
α·(β·x) = (β⋅α)·x
(for right Rpseudomodules)
Therefore, the law of action may be written more suggestively as ·:M×R→M
instead. 
right Rpseudomodule 


 R is a ring.
 n ∀x∈M
1·x = x

left Rmodule ⊂
left Rpseudomodule 
RMod 

 R is a ring.
 n ∀x∈M
1·x = x

right Rmodule ⊂
right Rpseudomodule 


R is a field 
left Rvector space ⊂
left Rmodule 
RVec 

R is a field 
right Rvector space ⊂
right Rmodule 

By an abuse of language, left Rmodules are
simply called Rmodules, and left Rvector spaces are simply called Rvector
spaces. The theory of pseudomodules can be reduced to that of modules by
adjoining a unit element. Left Rmodules are commutative groups with operators
in R and left (R,⋅)sets that also satisfy the second distributive relation.
Homomorphisms of algebraic structures with a law of composition and a
law of action
Let (M,+,·), (M',+,·) be
Rmodules/Rvector spaces, and φ:M→M' a map. 

If 

Then φ is a 
m 
 φ:(R,+)→(R',+) is a homomorphism of groups
 ∀α∈R∀x∈M φ(α·x)
= α·φ(x)

Rlinear 
homomorphism of ... 
Examples
 A pseudoring R is an Rmodule with ordinary ring multiplication;
(twosided) ideals I⊴R are the (left and right) subRmodule
thereof.
Be aware that proper ideals I⊲R (i.e. I⊴R ∧ I≠R) are only subpseudorings
and no unital subrings since 1∉I.
 Ab≅ZMod.
 If M and M' are Rmodules, Hom_{R}(M,M') := {φ:M→M' ¦ φ is
Rlinear} is an Rmodule with pointwise addition and scalar multiplication.
 M^{⌄} := M^{*} := Hom_{R}(M,R) is called the dual
module to M. If M is free with base B, then M^{*} is free with
the dual base B^{*}:={b^{*}:B→R;
b'↦δ_{bb'}
¦ b∈B}.
 R^{(X)} = ⊕_{x∈X} R = {∑_{x∈X}
a_{x}·x ¦
∀x∈X a_{x}∈R
∧ a_{x}=0 p.t. x∈X}
is the free Rmodule with basis X (in contrast to algebras
of a magma, X is only a set).
⇔ X is Rlinear independent and generates the module (i.e. =〈X〉).
Algebras
Algebraic structures with two laws of composition and a law of action
Let R be a commutative ring, E a set. 

If 
Then E is a 
C 

 E is an Rmodule
 ⋅:E×E→E is a law of composition that is bilinear,
i.e.
 (l^{2})=(d)
(x+y)⋅z = (x⋅z) + (y⋅z)
x⋅(y+z) = (x⋅y) + (x⋅z)

m^{2}
(α·x)⋅y = α·(x⋅y)
x⋅(α·y) = α·(x⋅y)

Ralgebra 
RAlg 
a 
⋅:E×E→E is associative 
associative Ralgebra ⊂
Ralgebra 
RAlgas 
n 
⋅:E×E→E has a neutral element 
unital Ralgebra ⊂
Ralgebra 
RAlgu 
c 
⋅:E×E→E is commutative 
commutative Ralgebra ⊂
Ralgebra 
RcAlg 
Note
In case of multiplication the sign ⋅ is often
omitted.
Alternative characterizations of associative algebras
Let R be a commutative ring, E a set. 

If 
Then E is a 

 E is an Rmodule
 E is a pseudoring
 (m) R⋅E ⊆ Z(E), i.e.
(α·x)⋅(β·y) = (α β)·(x⋅y)

associative Ralgebra 
ι 
ι:R→E is a homomorphism of rings.
(setting α·x := ι(α)⋅x, and the Ralgebra E inherits unital from ι) 
associative Ralgebra 
Note
The converse of the last characterization (ι) is also true for unital and
associative algebras, since given such an Ralgebra, ι:R→E;α↦α·1 is a
unital homomorphism of rings. Homomorphisms of Ralgebras are the Rlinear
homomorphisms of rings:
Homomorphisms of algebraic structures with two laws of composition
and a law of action
Let (E,+,·,⋅), (E',+,·,⋅) be
Ralgebras/unital Ralgebras, and φ:E→E' a map. 

If 

Then φ is a 
m 
 φ:(E,+,·)→(E',+,·) is a homomorphism of Rmodules
 ∀x,y∈E φ(x⋅y) = φ(x)⋅φ(y)

morph 
homomorphism of Ralgebras 
1 
φ(1) = 1 
unital 
unital homomorphism ⊂
homomorphism 
Examples
 A commutative pseudoring R is a commutative associative Ralgebra.
 A commutative ring R with 1 is a unital and commutative associative
Ralgebra.
 A ring extension R'≥R is an associative Ralgebra.
 If I⊴R is an ideal, R/I is an Ralgebra.
 ZAlgas≅PseudoRing,
ZAlgasu≅Rng1,
ZAlgascu≅cRng1.
 R[S] := R^{(S)} = ⊕_{s∈S} R = {∑_{s}_{∈S}
a_{s}·ι(s) ¦ ∀s∈S a_{s}∈R ∧ a_{s}=0 p.t.
s∈S} is the Ralgebra of the magma S over R
(being associative, unital, or commutative if and only if S is), by ι(s)⋅ι(t)
:= ι(s⋅t) from the canonical injection ι:S↪R^{(S)}. Therefore
by (d), the multiplication is the convolution
(∑_{s∈S} α_{s}·ι(s))⋅(∑_{s∈S} β_{s}·ι(s))
= ∑_{s∈S} (∑_{t⋅u=s} α_{t} β_{u})·ι(s)
Especially, there are
 the free Ralgebra Lib_{R}(I) := R^{(M(I))}.
 the free associative Ralgebra Libas_{R}(I) := R^{(Mo(I))
}≅ T(R^{(I)})
being isomorph to the tensor algebra of a free Rmodule with a basis (of
indexing set) I. (It also is a quotient of Lib_{R}(I)).
 the free commutative and associative Ralgebra Libasc_{R}(I)
:= R^{(N(I))} =: R[(X_{i})_{i∈I}]
≅ S(R^{(I)})
being isomorph to the symmetric algebra of a free Rmodule with a basis
(of indexing set) I. (It also is a quotient of Lib_{R}(I)).
The algebra R^{(S)} of a magma S over R is the presenting object of
the presentable functor
RAlg 
→ 
Ens 
E 
↦ 
Mor_{Mag}(S,(E,⋅))

Therefore it enjoys the following universal mapping property
∀φ:S→(E,⋅) homomorphism of magmas/monoids into an Ralgebra E
∃!Φ:R^{(S)}→E Ralgebra/unital Ralgebra homomorphism with Φ_{S}=φ
If R is a graded ring with graduation Δ, then a homomorphism deg:S→Δ
defines a graded Ralgebra structure on R^{(S)} by setting deg(α·ι(s))
:= deg(α)+deg(s).
 R^{X} = ∏_{x}_{∈X} R is a commutative
associative Ralgebra (for a commutative ring R and a set X), but is
distinct from the total algebra.
 The tensor product E⊗_{R}E' is a coproduct in the category of
Ralgebras (only RAlgu) per
(a⊗b)⋅(a'⊗b') :=a a'⊗b b'. It inherits (a),(n),(c) from E and
E', whence the tensor product is also a coproduct in RAlgasu,
RAlgascu etc.
 By adjoining a unit element, an Ralgebra E can be transformed and
extended to a unital Ralgebra, which is just as associative or commutative
as E.
 (End_{A}(M),+,∘) is an associative Ralgebra if M is a right
Amodule over an associative Ralgebra A.
Quotient Structures
Provided one of the following (equivalent) conditions is satisfied
 Let φ:M→M be a homomorphism on M and x~y :⇔ φ(x)=φ(y) the
equivalence relation φ induces on M.
 Let ~ ⊆ M×M be a congruence relation on M, i.e. an
equivalence relation with which all operators ⋆∈Σ of any arity n,
occurring in the Σalgebra M, are compatible, i.e. ∀x_{1},…,x_{n}∈M
∀y_{1},…,y_{n}∈M ((∀i x_{i}~y_{i})
⇒ ⋆(x_{1},…,x_{n}) ~ ⋆(y_{1},…,y_{n}))
Then the quotient M/~ := M̄ := {x̄ ¦ x∈M} with
x̄ := ^{1}f({f(x)}) = {y∈M ¦ x~y} for x∈M has the same
algebraic structure as M. If M is a magma, semigroup, monoid, group, Abelian
group, ring, Rmodule, or vector space, then M/~ is as well. The laws of
composition ⋆ and laws of action · on M/~ are obtained by passing to the
quotient:
x̄⋆ȳ := x⋆y
α·x̄ := α·x
Since ~ is an equivalence on M, M/~ is a partition of M
∀x̄,ȳ∈M
(x̄∩ȳ=∅ xor x̄=ȳ)Although conversely, every partition of M
defines an equivalence relation, that relation does not need to be a congruence
compatible with the algebraic structure of M.
Note that congruence relations ~ on M often correspond bijectively to special
substructures of M, and to (left or right) cosets .... . Examples are normal
subgroups in groups, subgroups in Abelian groups, ideals (i.e. subRmodules) in
rings, submodules in Rmodules, ideals (i.e. subRmodules and ideals
of the ring structure) in Ralgebras (which in RAlgasu
are identical with the ideals of the ring structure). For example, if (G,⋅) is
a group with a normal subgroup N⊴G, then there is a homomorphism φ with N=Ker(φ)
inducing a congruence relation ~, and
G/N := G/~ = {g⋅N:={g⋅n
¦ n∈N} ¦ g∈G} = {N⋅g:={n⋅g ¦ n∈N} ¦ g∈G}equals
the set of left cosets or the set of right cosets. Although the left
and right cosets are also defined for ordinary subgroups H≤G instead of N,
they do not coincide with any equivalence classes of a quotient group in the
general case. In either case, all cosets have the same cardinality and [G:H]
:= number of left cosets = number of right cosets (= G/H if H⊴G).
is called the index of the subgroup H≤G. It satisfies G =
[G:H]⋅H
Generating Systems
Let E have the algebraic structure T, and X⊆E
be a subset.
 generated
 〈X〉 := ⋂_{X⊆S≤E} S ≤ E
is the smallest stable subT of E containing
X. It is called the subT generated by X.
 For Rmodules E, an alternative characterization by linear combinations of
X⊆E is
〈X〉 = R·X := 〈{r·x ¦ r∈R ∧ x∈X}〉
= R^{(X)} (= {∑_{x∈X}
a_{x}·x ¦
∀x∈X a_{x}∈R
∧ a_{x}=0 p.t. x∈X})
 For ideals I⊴R (which are defined as subRmodules of R), the usual
notation is (X) instead of 〈X〉, and we have the same alternative
characterizations
(X) := ⋂_{X⊆S}_{⊴}_{E} S = X⋅R = R^{(X)}
(= {∑_{x∈X} a_{x}⋅x ¦ ∀x∈X a_{x}∈R
∧ a_{x}=0 p.t. x∈X})
 For ring extensions S/R (resp. commutative, unital associative Ralgebras
S), and a subset A⊆S we have an alternative characterization
R[A] (:= R[〈A〉_{Mag}]) = 〈R∪A〉 = ⋃_{E⊆A
fin} R[E] (= {f(α_{1},…,α_{n}) ¦ n∈N
∧ f∈R[X_{1},…,X_{n}] ∧ ∀i α_{i}∈A})
 For field extensions L/K, and a subset A⊆L we have an alternative
characterization
K(A) := Quot(K[A]) = 〈K∪A〉 = ⋃_{E⊆A fin} K(E) = ({f(α_{1},…,α_{n})
/ g(α_{1},…,α_{n}) ¦ n∈N ∧ f,g∈K[X_{1},…,X_{n}]
∧ ∀i α_{i}∈A ∧ g(α_{1},…,α_{n})≠0})
(= K[A] if all α∈A are algebraic over K)
The word algebra comes from the book "Kitāb alǧabr
walmuquābala" by Abū'
Abdallah Muhammad ibn Mūsā alMaǧūsī
AlHwārizmī alChoresmi
(787ca.850)