A Commutative Algebra proof of parallel universes.

The following proof will show that the Big Bang created many values for the Real number one and that this universe has exactly one value for the Real number one and therefore the other parallel universes have other values for the Real number one.

Here are the three basic laws of equality for any commutative ring R [1].

Reflexive law: a = a.

Symmetric law: If a = b, then b = a.

Transitive law: If a = b and b = c, then a = c, valid for all a, b, and c.

For all a in R, 1 * a = a.

Proof. For all a in R:

1) 1 * a = a * 1

2) a * 1 = a

3) 1 * a = a.

If u in R has the property that a * u = a for all a in R, then u = 1.

Proof. Since a * u = a holds for all a, it holds if a is 1.

1) 1 * u = 1

2) 1 = 1 * u

3) u = u * 1

4) u = 1.

With the above proofs we can now create a less simplified form of the three basic laws of equality for any commutative ring R for any single value of u.

Reflexive law: a * u = a * u.

Symmetric law: If a * u = b * u, then b * u = a * u.

Transitive law: If a * u = b * u and b * u = c * u, then a * u = c * u, valid for all a, b, and c.

We now propose the Meta-commutative ring R_u, where when u = 1 then it denotes the normal commutative ring R of this universe. For meta-commutative ring R_u, u ≠ 0.

Here are the three basic laws of equality of the meta-commutative ring R_u.

Reflexive law: a * u = a * u.

Symmetric law: If a * u = b * u, then b * u = a * u.

Transitive law: If a * u = b * u and b * u = c * u, then a * u = c * u, valid for all a, b, and c.

Each separate R_u defines a separate commutative ring.

Proof. If R₁ exists then R_m exists along with R_n. n ≠ m ≠ 1.

1) For all R_u, then a * u = a * u.

2) If R_n is not separate from R_m then (a * n = a * n) = (a * m = a * m).

3) But they are not equal, therefore R_n is a separate Commutative Ring from R_m.

References

1. Birkhoff, Garrett, and MacLane, Saunders. “A Survey Of Modern Algebra”. (1997). A. K. Peters, Wellesley, Massachusetts.