Equivalence of Definitions of Unit of Ring

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

Proof
Let $\struct {R, +, \circ}$ be a ring with unity.

$(1)$ implies $(2)$
Let $x \in R$ be a unit of $\struct {R, +, \circ}$ by definition 1.

Then by definition:
 * $\exists y \in R: x \circ y = 1_R = y \circ x$

That is, by definition of divisor:


 * $x \divides 1_R$

Thus $x$ is a unit of $\struct {R, +, \circ}$ by definition 2.

$(2)$ implies $(1)$
Let $x \in R$ be a unit of $\struct {R, +, \circ}$ by definition 2.

Then by definition:
 * $x \divides 1_R$

By definition of divisor:
 * $\exists t \in R: 1_R = t \circ x$

Thus $x$ is a unit of $\struct {R, +, \circ}$ by definition 1.