Finite Local Rings of Length 4

This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field Fpm, where p is a prime number. Such rings have an order of p4m elements. The current paper provides the structure and classification, up to isomorphism, of local rings consisting of p4m elements. We also give the exact number of non-isomorphic classes of these rings with fixed invariants p,n,m,k. In particular, we have listed all finite local rings of 4-length and of order p8 and 256.