In this paper, we aresupposed to introduce the definitions of n-fold commutative, andimplicative hyper K-ideals. These definitions are thegeneralizations of the definitions of commutative, andimplicative hyper K-ideals, respectively, which have been definedin [12]. Then we obtain some related results. In particular wedetermine the relationships between n-fold implicative hyperK-ideal and n-fol...

Given a minimal model of an elliptic curve, $E/K$, over finite extension, $K$, ${\mathbb Q}_{p}$ for any rational prime, $p$, and point $P \in E(K)$ infinite order, we determine precisely $\min \left( v \phi_{n}(P) \right), \psi_{n}^{2}(P) \right) \right)$, where $v$ is normalised valuation on $K$ $\phi_{n}(P)$ $\psi_{n}(P)$ are polynomials arising from multiplication by $n$ this the curve.