SPECIAL WEAK PROPERTIES OF GENERALIZED POWER SERIES RINGS

Abstract

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