0/0 exists in the 0 ring

{0} is a ring under multiplication. 0 is the multiplicative identity and additive identity.

we have 0 x 0 = 0
so 0/0 = 0 and everythign works fine!

A RING IS AN ABELIAN GROUP UNDER ADDITION THAT IS ALSO CLOSED, ASSOCIATIVE, AND DISTRIBUTIVE UNDER MULTIPLICATION.  {0} IS A RING

AN INTEGRAL DOMAIN IS A COMMUTATIVE (UNDER MULT.) RING, WITH UNITY AND NO ZERO DIVISORS.
CONSEQUENTLY, {0} IS ALSO AN INTEGRAL DOMAIN, WHERE 0 ACTS AS UNITY.  NOTE: THERE ARE NO NON-ZERO ELEMENTS.

ALL FINITE INTEGRAL DOMAINS ARE FIELDS, SO {0} IS A FIELD AS WELL.