Special Weak Properties of Generalized Power Series Rings

Abstract

Abstract. 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 ∈ R | xa ∈ nil(R) for all x ∈ X}, which is called a weak annihilator of X in R. A ring R is called weak zip provided that for any subset, and a ring R is called weak symmetric if abc ∈ nil(R) ⇒ acb ∈ nil(R) for all a, b, c ∈ R. It is shown that a generalized power series ring [[R S,≤ ]] 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,≤ ]] in terms of all weak associated primes of R in a very straightforward way.

A Note on Skew Generalized Power Serieswise Reversible Property

“…If R is a NI-ring, it is evident that Nr R (X) forms an ideal of R. Additionally, if R is reduced, then we have r R (X) = Nr R (X) = l R (X) = Nl R (X), and more information and findings on weak annihilators can be found in [14].…”