Step 1: Understanding the lac operon.
The lac operon is a genetic system found in bacteria, such as E. coli, which controls the metabolism of lactose. It consists of three structural genes: z, y, and a. The z-gene specifically codes for the enzyme $\beta$-galactosidase, which breaks down lactose into glucose and galactose.
Step 2: Analyze the options.
- z-gene: This gene codes for $\beta$-galactosidase, the enzyme responsible for lactose metabolism.
- a-gene: This gene codes for an enzyme called transacetylase, not $\beta$-galactosidase.
- y-gene: This gene codes for permease, which facilitates the transport of lactose into the cell but not the breakdown of lactose.
- i-gene: This gene codes for the repressor protein that regulates the lac operon but does not directly code for $\beta$-galactosidase.
Step 3: Conclusion.
The correct gene responsible for coding $\beta$-galactosidase in the lac operon is the z-gene.
Final Answer:
\[
\boxed{z\text{-}gene}
\]