## Overview

Three levels of abstraction in algebra are

1. concrete algebra: specific elements of a specific set with precisely defined operations.
2. abstract algebra / axiomatic algebra: general algebraic structures with operations satisfying postulates, instead of specific sets.
3. 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.
sub-T
U≤E :⇔ U⊆E ∧ (U,⋅|U×U) has itself an algebraic structure T.
U is called a sub-T then.
isomorphism
 If Let E,F have the algebraic structure T, and φ:E→F be a map. iso ∃ψ:F→E homomorphism with φ∘ψ=idF ∧ ψ∘φ=idE 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

 If Then M is a C Let M be a set. 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.

 If Then φ is a Let (M,⋅), (N,∗) be magmas/semigroups/monoids/groups/Abelian groups, and φ:M→N a map. 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):=-1φ({0})⊴M are sub-T (and Ker(φ) even is a normal subgroup resp. an ideal for groups resp. rings or algebras). Also for N'≤N, Ker(φ)⊆-1φ(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

1. 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.
2. For every set X there is the free semigroup (of words=lists) Fa(X) := X+ := ⋃n∈N\{0} Xn.
3. For every set X there is the free monoid (of words) Mo(X) := X* := ⋃n∈N Xn.
4. For every set X there is the free commutative monoid N(X).
5. For every set X there is the free group F(X) with basis X where
∀y∈F(X) ∃!x1ε1⋅...⋅xnεn = y with xi∈X ∧ εi∈{-1,1} ∧ xiεi≠xi+1i+1.
6. AbZ-Mod.
7. The Cartesian product ∏i∈I Ai := {(ai)i∈I ¦ ai∈Ai} with component-wise laws (requiring AxCh) is the product in Ens,Mag,Sgp,Mon,Gr,Ab,Rng,Rng1,cRng,R-Mod,K-Vec,R-Alg,R-Algas,R-Alg1,... (but Fld has no product).
Especially AI := ∏i∈I A.
8. The direct sum i∈I Ai := {(ai)i∈I ∈ ∏i∈I Ai ¦ ai=1 p.t. i∈I} with component-wise laws is the coproduct in cMon, Mon??,Ab,R-Mod,K-Vec,..., but that is not the coproduct in Gr nor in R-Alg.
Especially A(I) := i∈I A.
9. (XX,∘) is a monoid for any set X, and SX := Perm(X) := (XX)× ≤ X is the symmetric group. If M is a magma, then Aut(M)≤SM.
10. If M is a magma and C a commutative T, then Hom(M,C)≤CM is a commutative sub-T. (T is one of Sgp,Mon,Gr,Ab,R-Mod,K-Vec, R is a commutative? ring ).

### Groups and actions

 If Then E is a C Let Ω,E be sets. s ·:Ω→EE 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→EE is an action of Ω on E. s ^ is distributive over ⋅, i.e. ∀α∈Ω ∀x,y∈E (x⋅y)α = (xα) ⋅ (yα). ∀α∈Ω ^α is an endomorphism of E. group with operators ⊂ group∩"set with action" GpOp-Ω
##### Note

Groups are identified with groups with operators in ∅. By an abuse of language, laws of left actions are simply called laws of actions.

 If Then φ is a Let M,N be groups with operators in Ω, and φ:M→N a map. m ∀α∈Ω ∀x∈M φ(xα) = φ(x)α morph homomorphism of groups with operators ⊂ homomorphism of groups
 If Then E is a Let G be a group, E a set. (G,⋅) is a group ·:G→EE 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 G-set G-Ens (G,⋅) is a group ·:G→EE is a homomorphism of monoids into the opposite monoid of EE (right operation), i.e. ∀α,β∈G ∀x∈E (x·α)·β = x·(α⋅β). ∀x∈E x·1 = x. right G-set
##### 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→SE.

 If Then φ is a Let E,E' be G-sets, and φ:E→E' a map. m ∀α∈G ∀x∈E φ(α·x) = α·φ(x) morph homomorphism of G-sets

#### Examples

1. _·x is a homomorphism of G-sets (or G-morphism) 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.
Gx := {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,Gx=G. Fixgroups and orbits are related by |G·x| = [G:Gx].

### Rings

 If Then R is a C Let R be a set. (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) pseudo-ring n (R,⋅) is a monoid ring with 1 ⊂ pseudo-ring 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 pseudo-rings and true rings with 1. Some authors also refer to a ring when they think of what we called a pseudo-ring, and will always speak of a ring with 1 if they talk about true rings with 1.

A pseudo-ring 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
 If Then φ is a Let (R,+,⋅), (R',+,⋅) be pseudo-rings/rings with 1/commutative rings/skew fields/fields, and φ:R→R' a map. m φ:(R,+)→(R',+) is a homomorphism of groups. φ:(R,⋅)→(R',⋅) is a homomorphism of semigroups. morph homomorphism of pseudo-rings 1 φ(1) = 1, i.e. φ:(R,⋅)→(R',⋅) is a homomorphism of monoids. unital homomorphism of ... ⊂ homomorphism of pseudo-ring
##### Note

Rng1Grp; (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 skew-field R are injective. Also compare with homomorphisms of groups to see that if R and R' are skew-fields, then for homomorphisms of rings the property (1) follows from (m).

#### Examples

1. Z-AlgasPseudoRing, Z-AlgasuRng1, Z-AlgascucRng1.
2. The tensor product R⊗ZR' is a coproduct in the category of commutative rings with 1. But no coproduct exists in the category of rings???

### Modules

 If Then M is a C Let R be a (pseudo-)ring, M a set. (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 R-pseudomodules) left R-pseudomodule The same as for left R-modules, except that a ∀α,β∈A ∀x∈M α·(β·x) = (β⋅α)·x (for right R-pseudomodules) Therefore, the law of action may be written more suggestively as ·:M×R→M instead. right R-pseudomodule R is a ring. n ∀x∈M 1·x = x left R-module ⊂ left R-pseudomodule R-Mod R is a ring. n ∀x∈M 1·x = x right R-module ⊂ right R-pseudomodule R is a field left R-vector space ⊂ left R-module R-Vec R is a field right R-vector space ⊂ right R-module

By an abuse of language, left R-modules are simply called R-modules, and left R-vector spaces are simply called R-vector spaces. The theory of pseudomodules can be reduced to that of modules by adjoining a unit element. Left R-modules are commutative groups with operators in R and left (R,⋅)-sets that also satisfy the second distributive relation.

 If Then φ is a Let (M,+,·), (M',+,·) be R-modules/R-vector spaces, and φ:M→M' a map. m φ:(R,+)→(R',+) is a homomorphism of groups ∀α∈R∀x∈M φ(α·x) = α·φ(x) R-linear homomorphism of ...

#### Examples

1. A pseudo-ring R is an R-module with ordinary ring multiplication; (two-sided) ideals I⊴R are the (left and right) sub-R-module thereof.
Be aware that proper ideals I⊲R (i.e. I⊴R ∧ I≠R) are only sub-pseudo-rings and no unital sub-rings since 1∉I.
2. AbZ-Mod.
3. If M and M' are R-modules, HomR(M,M') := {φ:M→M' ¦ φ is R-linear} is an R-module with pointwise addition and scalar multiplication.
4. M := M* := HomR(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'↦δbb' ¦ b∈B}.
5. R(X) = x∈X R = {∑x∈X ax·x ¦ ∀x∈X ax∈R ∧ ax=0 p.t. x∈X} is the free R-module with basis X (in contrast to algebras of a magma, X is only a set).
⇔ X is R-linear independent and generates the module (i.e. =〈X〉).

### Algebras

 If Then E is a C Let R be a commutative ring, E a set. E is an R-module ⋅:E×E→E is a law of composition that is bilinear, i.e. (l2)=(d) (x+y)⋅z = (x⋅z) + (y⋅z) x⋅(y+z) = (x⋅y) + (x⋅z) m2 (α·x)⋅y = α·(x⋅y) x⋅(α·y) = α·(x⋅y) R-algebra R-Alg a ⋅:E×E→E is associative associative R-algebra ⊂ R-algebra R-Algas n ⋅:E×E→E has a neutral element unital R-algebra ⊂ R-algebra R-Algu c ⋅:E×E→E is commutative commutative R-algebra ⊂ R-algebra R-cAlg
##### Note

In case of multiplication the sign ⋅ is often omitted.

 If Then E is a Let R be a commutative ring, E a set. E is an R-module E is a pseudo-ring (m) R⋅E ⊆ Z(E), i.e. (α·x)⋅(β·y) = (α β)·(x⋅y) associative R-algebra ι ι:R→E is a homomorphism of rings. (setting α·x := ι(α)⋅x, and the R-algebra E inherits unital from ι) associative R-algebra
##### Note

The converse of the last characterization (ι) is also true for unital and associative algebras, since given such an R-algebra, ι:R→E;α↦α·1 is a unital homomorphism of rings. Homomorphisms of R-algebras are the R-linear homomorphisms of rings:

 If Then φ is a Let (E,+,·,⋅), (E',+,·,⋅) be R-algebras/unital R-algebras, and φ:E→E' a map. m φ:(E,+,·)→(E',+,·) is a homomorphism of R-modules ∀x,y∈E φ(x⋅y) = φ(x)⋅φ(y) morph homomorphism of R-algebras 1 φ(1) = 1 unital unital homomorphism ⊂ homomorphism

#### Examples

1. A commutative pseudo-ring R is a commutative associative R-algebra.
2. A commutative ring R with 1 is a unital and commutative associative R-algebra.
3. A ring extension R'≥R is an associative R-algebra.
4. If I⊴R is an ideal, R/I is an R-algebra.
5. Z-AlgasPseudoRing, Z-AlgasuRng1, Z-AlgascucRng1.
6. R[S] := R(S) = s∈S R = {∑s∈S as·ι(s) ¦ ∀s∈S as∈R ∧ as=0 p.t. s∈S} is the R-algebra 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 αs·ι(s))⋅(∑s∈S βs·ι(s)) = ∑s∈S (∑t⋅u=s αt βu)·ι(s)
Especially, there are
• the free R-algebra LibR(I) := R(M(I)).
• the free associative R-algebra LibasR(I) := R(Mo(I)) T(R(I)) being isomorph to the tensor algebra of a free R-module with a basis (of indexing set) I. (It also is a quotient of LibR(I)).
• the free commutative and associative R-algebra LibascR(I) := R(N(I)) =: R[(Xi)i∈I] ≅ S(R(I)) being isomorph to the symmetric algebra of a free R-module with a basis (of indexing set) I. (It also is a quotient of LibR(I)).
The algebra R(S) of a magma S over R is the presenting object of the presentable functor
 R-Alg → Ens E ↦ MorMag(S,(E,⋅))
Therefore it enjoys the following universal mapping property
∀φ:S→(E,⋅) homomorphism of magmas/monoids into an R-algebra E
∃!Φ:R(S)→E R-algebra/unital R-algebra homomorphism with Φ|S
If R is a graded ring with graduation Δ, then a homomorphism deg:S→Δ defines a graded R-algebra structure on R(S) by setting deg(α·ι(s)) := deg(α)+deg(s).
7. RX = ∏x∈X R is a commutative associative R-algebra (for a commutative ring R and a set X), but is distinct from the total algebra.
8. The tensor product E⊗RE' is a coproduct in the category of R-algebras (only R-Algu) 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 R-Algasu, R-Algascu etc.
9. By adjoining a unit element, an R-algebra E can be transformed and extended to a unital R-algebra, which is just as associative or commutative as E.
10. (EndA(M),+,∘) is an associative R-algebra if M is a right A-module over an associative R-algebra A.

## Quotient Structures

Provided one of the following (equivalent) conditions is satisfied

1. Let φ:M→M be a homomorphism on M and x~y :⇔ φ(x)=φ(y) the equivalence relation φ induces on M.
2. 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 Σ-algebra M, are compatible, i.e.
∀x1,…,xn∈M ∀y1,…,yn∈M ((∀i xi~yi) ⇒ ⋆(x1,…,xn) ~ ⋆(y1,…,yn))
Then the quotient
M/~ := M̄ := {x̄ ¦ x∈M} with
x̄ := -1f({f(x)}) = {y∈M ¦ x~y} for x∈M
has the same algebraic structure as M. If M is a magma, semi-group, monoid, group, Abelian group, ring, R-module, 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. sub-R-modules) in rings, submodules in R-modules, ideals (i.e. sub-R-modules and ideals of the ring structure) in R-algebras (which in R-Algasu 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 sub-T of E containing X. It is called the sub-T generated by X.
• For R-modules E, an alternative characterization by linear combinations of X⊆E is
〈X〉 = R·X := 〈{r·x ¦ r∈R ∧ x∈X}〉 = R(X) (= {∑x∈X ax·x ¦ ∀x∈X ax∈R ∧ ax=0 p.t. x∈X})
• For ideals I⊴R (which are defined as sub-R-modules of R), the usual notation is (X) instead of 〈X〉, and we have the same alternative characterizations
(X) := ⋂X⊆SE S = X⋅R = R(X) (= {∑x∈X ax⋅x ¦ ∀x∈X ax∈R ∧ ax=0 p.t. x∈X})
• For ring extensions S/R (resp. commutative, unital associative R-algebras S), and a subset A⊆S we have an alternative characterization
R[A] (:= R[〈A〉Mag]) = 〈R∪A〉 = ⋃E⊆A fin R[E] (= {f(α1,…,αn) ¦ n∈N ∧ f∈R[X1,…,Xn] ∧ ∀i αi∈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(α1,…,αn) / g(α1,…,αn) ¦ n∈N ∧ f,g∈K[X1,…,Xn] ∧ ∀i αi∈A ∧ g(α1,…,αn)≠0})
(= K[A] if all α∈A are algebraic over K)

The word algebra comes from the book "Kitāb al-ǧabr wa-l-muquābala" by Abū' Abdallah Muhammad ibn Mūsā al-Maǧūsī Al-Hwārizmī al-Choresmi (787-ca.850)