# Categories of Elements of Ring

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## 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.

$\blacksquare$