The Birth of Abstract Algebra

The birth of Abstract Algebra :
Galois “Group Theory” – >
Dedekind “Field Theory” ->
Noether (Ring / Ideal)
-> Noether Axiomatic Abstract Algebra

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Mathematician Emmy Noether and her Physics “Energy Conservation” Theorems

The Abstract Algebra was made “axiomatic” by Noether, whose student Van der Waerden compiled her lecture notes into the first Abstract Algebra textbook in the world 《Modern Algebra》.

“Noether Ring” is the core of Ring Theory pioneered by Noether.

https://plus.maths.org/content/noether

Lord of the Ring

Lord of the “Ring”:
The term Ring first introduced by David Hilbert (1862-1943) for Z and Polynomial.
The fully abstract axiomatic theory of commutative rings by his student Emmy Noether in her paper “Ideal Theory in Rings” @1921.

eg. 3 Classical Rings:
1. Matrices over Field
2. Integer Z
3. Polynomial over Field.

Ring Confusions
Assume all Rings with 1 for * operation.

Ring has operation + forms an Abelian group, operation * forms a semi group (Close, Associative).

1) Ever ask why must be Abelian + group ?
Apply Distributive Axioms below:
(a+b).(1+1) = a.(1+1) + b.(1+1)
= a + a + b + b …[1]

Or,
(a+b).(1+1) = (a+b).1 + (a+b).1
= a + b + a + b …[2]

[1]=[2]:
a + (a + b) + b = a + (b + a) +b
=> a + b = b + a

Therefore, + must be Abelian in order for Ring’s * to comply with distributive axiom wrt +.

2). Subring
Z/6Z ={0,1,2,3,4,5}
3.4=0 => 3, 4 zero divisor

has subrings: {0,2,4},{0,3}

3). Identity 1 and Units of Ring

Z/6Z has identity 1
but 2 subrings do not have 1 as identity.
subrings {0,2,4}:
0.4=0
2.4=2,
4.4=4 => identity is 4
4 is also a unit.

Units: Ring R with 1.
∀a ∈ R ∃b ∈ R s.t.
a.b=b.a = 1
=> a is unit
and b its inverse a^-1

Z/6Z: identity for * is 1
5.5 = 1
5 is Unit besides 1 which is also unit. (1.1=1)

Noether Theorem: Symmetry

Symmetry (hence Group) explains :
1. Conservation of Energy;
2. Conservation of Angular Momentum;
3. Periodic Table;
4. Laws of Thermodynamic.

Emmy Noether Theorem (1918): Conservation Laws owes to Symmetry :
1. In Linear motion
=> Conservation of Momentum

2. In Angular movement
=> Conservation of Angular Momentum

3. In Time
=> Conservation of Energy