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)