A $\beta$-lactam ring is a four-membered lactam ring in the structure of certain antibiotics, which inhibits bacterial cell wall synthesis. Amoxicillin, a penicillin-class antibiotic, contains a $\beta$-lactam ring in its structure, making it effective against a wide range of bacteria. Erythromycin (a macrolide), Vancomycin (a glycopeptide), and Ciprofloxacin (a fluoroquinolone) do not have a $\beta$-lactam ring in their structures.
Thus, the correct answer is Amoxicillin.