Ring is not Empty

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Theorem

A ring can not be empty.


Proof

In a ring $\left({R, +, \circ}\right)$, $\left({R, +}\right)$ forms a group.

From Group is not Empty, the group $\left({R, +}\right)$ contains at least the identity, so can not be empty.

So every ring $\left({R, +, \circ}\right)$ contains at least the identity for ring addition.

$\blacksquare$


Sources