Examples proving the independence of field axioms

Question

I wanted to find examples from the field where one of the conditions of the field is not fulfilled every time, but the rest of the conditions are fulfilled. For example, "+" is not associative, but the rest of the fields are for exampel (z,+,.) does not have the condition of the .

Answer 1

The set $\{\text{true}, \text{false}\}$ with addition $\vee$ (the logical "or") and multiplication $\wedge$ (the logical "and") is an example. Only the condition that there must exist an additive inverse for every element is being harmed.

Answer 2

I think that you are looking for examples of rings that are not fields.

A classical one is the ring of integers $(\mathbb{Z},+,\cdot)$. Try to understand why this is not a field and come back if you have any doubts.

Another classical example is the ring of quaternions $(\mathbb{H},+,\cdot)$. A quaternion is a number of the form $$a_0 + a_1 i + a_2 j + a_3 k$$ where $a_0, a_1, a_2, a_3$ are real numbers, and $i, j, k$ are distinct imaginary units.

Given two quaternions $p := a_0 + a_1 i + a_2 j + a_3 k$ and $q := b_0 + b_1 i + b_2 j + b_3 k$, we define their sum as $$ p+q := (a_0 + b_0) + (a_1 + b_1) i + (a_2 + b_2) j + (a_3 + b_3) k.$$

The product of two quaternions is given by the unique associative and distributive binary operation satisfying $$i^2 = j^2 = k^2 = ijk = -1.$$

Note that multiplication is not commutative.

There is also another interesting example: the octonions.

It’s a good exercise to check that these are not fields.