Special weak properties of generalized power series rings
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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 de ne NR(X) = {a ∈ R｜ xa ∈ nil(R)for all x ∈ Xg, 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 NR(X) ⊆ nil(R),then there exists a nite subset Y ⊆ X such that NR(Y ) ⊆ nil(R), and a ring R is called weak symmetric if abc 2 nil(R) ) acb ∈ nil(R) for all a, b, c ∈ R. It is shown that a generalized power series ring [[RS,]] 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 [[RS,≤]] in terms of all weak associated primes of R in a very straightforward way.
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 de ne NR(X) = {a ∈ R｜ xa ∈ nil(R)for all x ∈ Xg, 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 NR(X) ⊆ nil(R),then there exists a nite subset Y ⊆ X such that NR(Y ) ⊆ nil(R), and a ring R is called weak symmetric if abc 2 nil(R) ) acb ∈ nil(R) for all a, b, c ∈ R. It is shown that a generalized power series ring [[RS,]] 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 [[RS,≤]] in terms of all weak associated primes of R in a very straightforward way.
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