Journal of the
Korean Mathematical Society JKMS

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 $N_R(X)\subseteq nil(R),$ 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.

Keywords: weak annihilator, weak associated prime, generalized power series

MSC numbers: 16W60, 16S36