Categories of Elements of Ring

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

The elements of $R$ are partitioned into three classes:
 * $(1): \quad$ the zero
 * $(2): \quad$ the units
 * $(3): \quad$ the proper elements.

Proof
By definition, a proper element is a non-zero element which has no product inverse.

Also by definition, a unit is an element which does have a product inverse.

Because $0 \circ x = 0$ there can be no $x \in R$ such that $0 \times x = 1$, and so $0$ is not a unit.

Hence the result.