This paper proposes the generalization of our previous work to the ring A n= F 3 d [X]/(X n).
All results found before in A 2, A 3 and A 4 [1],[2],[3] hold in A n; but the approach here is
clearly different, and has given more interesting results, specially when 3 does not divide# E
a 0, b 0 1; the elliptic curve over the ring A n is a direct sum of the elliptic curve over the field
F 3 d and, unexpectedly its own subgroup of elements with the third projective coordinate
not invertible, instead of F 3 dn as it was thought in the earlier works. Other results are …

… In section 2 we will define the ring An and establish some useful results which are necessary
for the rest of this article, and in section 3 we will define: the elliptic curve over An and, explicitly
the group law over En … In this section we will define the elliptic curve over the ring An, classify
its elements and define explicitly the group law over it. … In this subsection we will classify
the elements of the elliptic curve into two types, depending on their projective coordinates, the
result is shown in the following proposition, which can be proven in the same way as in [3]. …