Zero of Inverse Completion of Integral Domain

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

Let $\struct {K, \circ}$ be the inverse completion of $\struct {D, \circ}$ as defined in Inverse Completion of Integral Domain Exists.

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

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

That is, any element of $K$ of the form $\dfrac {0_D} q$ acts as the zero of $K$.

Proof
Let us define $\eqclass {\tuple {a, b} } \ominus$ as in the Inverse Completion of Integral Domain Exists.

That is, $\eqclass {\tuple {a, b} } \ominus$ is an equivalence class of elements of $D \times D^*$ under the congruence relation $\ominus$.

$\ominus$ is the congruence relation defined on $D \times D^*$ by:
 * $\tuple {x_1, y_1} \ominus \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$

By Inverse Completion of Integral Domain Exists:
 * $\dfrac p q \equiv \eqclass {\tuple {p, q} } \ominus$

From Equality of Division Products, two elements $\dfrac a b, \dfrac c d$ of $K$ are equal :
 * $a \circ d = b \circ c$

This is consistent with the fact that two elements $\eqclass {\tuple {a, b} } \ominus, \eqclass {\tuple {c, d} } \ominus$ of $K$ are equal $a \circ d = b \circ c$.

Suppose $a = 0_D$ and $a \circ d = b \circ c$.

Hence:
 * $\eqclass {\tuple {0_D, b} } \ominus = \eqclass {\tuple {0_D, d} } \ominus$

Thus all elements of $K$ of the form $\eqclass {\tuple {0_D, k} } \ominus$ are equal, for all $k \in D^*$.

To emphasise the irrelevance of the $k$, we will abuse our notation and write:
 * $\eqclass {\tuple {0_D, k} } \ominus$

as
 * $\eqclass {0_D} \ominus$

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

Again abusing our notation, we will write:
 * $\eqclass {\tuple {a, b} } \ominus \circ \eqclass {\tuple {c, d} } \ominus$

to mean:
 * $\eqclass {\tuple {a \circ c, b \circ d} } \ominus$

So:

Hence:
 * $\eqclass {0_D} \ominus \circ \eqclass {\tuple {a, b} } \ominus = \eqclass {\tuple {a, b} } \ominus = \eqclass {\tuple {a, b} } \ominus \circ \eqclass {0_D} \ominus$

So $\eqclass {0_D} \ominus$ fulfils the role of a zero for $\tuple {K, \circ}$ as required.

Also we have that:

So $\eqclass {0_D} \ominus$ is idempotent.

It follows that $\eqclass {0_D} \ominus$ can be identified with $0_D$ from the mapping $\psi$ as defined in Construction of Inverse Completion.