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)

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