Set of Order 3 Vectors under Cross Product does not form Ring

Theorem

Let $S$ be the set of all vectors in a vector space of dimension $3$.

Let $\times$ denote the cross product operation.

Then the algebraic structure $\struct {S, +, \times}$ is not a ring.

Proof

For $\struct {S, +, \times}$ to be a ring, it is a necessary condition that $\struct {S, \times}$ is a semigroup.

For $\struct {S, \times}$ to be a semigroup, it is a necessary condition that $\times$ is associative on $S$.

However, from Vector Cross Product is not Associative, this is not the case here.

The result follows.

$\blacksquare$

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Some 'non-examples': $\text {(f)}$