Let $R$ be a ring and $nil(R)$ the set of all nilpotent elements of $R$. For a subset $X$ of a ring $R$, we define $N_R(X)={a{in}R{mid}xa{in}nil(R)$ for all $x{in}X$}, which is called a weak annihilator of $X$ in $R$. $A$ ring $R$ is called weak zip provided that for any subset $X$ of $R$, if $N_R(Y){subseteq}nil(R)$, then there exists a finite subset $Y{subseteq}X$ such that $N_R(Y){subseteq}nil(R)$, and a ring $R$ is called weak symmetric if $abc{in}nil(R){Rightarrow}acb{in}nil(R)$ for all a, b, $c{in}R$. It is shown that a generalized power series ring $[[R^{S,{leq}}]]$ is weak zip (resp. weak symmetric) if and only if $R$ is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring $[[R^{S,{leq}}]]$ in terms of all weak associated primes of $R$ in a very straightforward way.