Ring of Quaternions is Ring

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Theorem

The set $\mathbb H$ of quaternions forms a ring under the operations of addition and multiplication.


Proof

From Quaternion Addition forms Abelian Group‎, $\mathbb H$ forms an abelian group under quaternion addition.


From the definition it is clear that quaternion multiplication is closed.


We have from Matrix Form of Quaternion that quaternions can be expressed in matrix form.

From Quaternion Multiplication‎ we have that quaternion multiplication can be expressed in terms of matrix multiplication, which is associative.

So $\mathbb H$ forms a semigroup under quaternion multiplication.


Finally, we have that Matrix Multiplication Distributes over Matrix Addition.


So all the conditions are fulfilled for $\mathbb H$ to be a ring.

$\blacksquare$


Sources