The development of secure cryptographic protocols and the subsequent attack mechanisms have been placed in the existing literature with the utmost curiosity. While sophisticated quantum attacks bring a concern to the classical cryptographic protocols present in the applications that are used in common life, the necessity of developing post-quantum protocols is felt primarily. In the context of post-quantum cryptography, elliptic curve-base protocols are of special interest to the researchers. While the comprehensive study of elliptic curves over finite fields is well known, the extended study over finite rings is still missing. In this work, we generalize the study of Weierstrass elliptic curves over finite ring Z_n through classification. Several expressions to compute key factors in the study of elliptic curves are conferred. An all-around computational classification on the Weierstrass elliptic curves over Z_n for rigorous understanding is attached here.