Zero of Inverse Completion of Integral Domain

Theorem
Let $$\left({D, +, \circ}\right)$$ be an integral domain whose zero is $$0_D$$.

Let $$\left({K, \circ}\right)$$ be the inverse completion of $$\left({D, \circ}\right)$$ as defined in Inverse Completion of Integral Domain.

Let $$x \in K: x = \frac p q$$ such that $$p = 0_D$$.

Then $$x$$ is equal to the zero of $$K$$.

Proof
By the method of its construction, all elements of $$K$$ are of the form $$\frac p q$$ where $$p \in D$$ and $$q = D^*$$.

From Equality of Division Products, two elements $$\frac a b, \frac c d$$ of $$K$$ are equal iff $$a \circ d = b \circ c$$.

Suppose $$a = 0_D$$.

Then $$0_D \circ d = b \circ c = 0_D$$

But $$c \in D^*$$ so $$c \ne 0$$, which means $$b = 0$$

Thus all elements of $$K$$ of the form $$\frac 0 x$$ are equal, for all $$x \in D^*$$.

Next, by Product of Division Products, we have that $$\frac a b \circ \frac c d = \frac {a \circ b} {c \circ d}$$.

So $$\frac 0 x \circ \frac a b = \frac {0 \circ a} {x \circ b} = \frac 0 {x \circ b} = \frac {a \circ 0} {b \circ x} = \frac a b \circ \frac 0 x$$.

Also, $$\frac 0 x \circ \frac 0 x = \frac 0 {x \circ x}$$ and so $$\frac 0 x$$ is idempotent.

The result follows.