In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete fields K 1 and 2 with perfect residue fields, show that if n -th rings are isomorphic for each ? , then isometric isomorphic. More generally, there is depending only on ramification indices such homomorphism from ring to can be lifted a between valuation rings. Moreover...

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...