Null Ring iff Zero and Unity Coincide

Theorem
The null ring is the only ring in which the unity and zero coincide.

Proof
The single element of the null ring serves as an identity for both of the operations.

So, in this particular ring, the unity and the zero are the same element.

A non-null ring contains some non-zero element $a$.

Since $1_R \circ a = a \ne 0_R = 0_R \circ a$, then $1_R \ne 0_R$.

So if a ring is non-null, its unity cannot be zero.

So a ring is null iff its unity is also its zero.

Also see

 * Group with Zero Element is Trivial