On Weak Symmetric Rings

Semicommutative Property on Nilpotent Products

“…The nil-semicommutative property between R[x; σ, δ] and R is studied by Ouyang and Chen [21], when R is a (σ, δ)-compatible reversible ring. In this section, we continue to study for the nil-semicommutative property of R[x; σ, δ] and R[[x; σ]].…”

“…Following [21], for integers i, j with 0 ≤ i ≤ j, let f j i ∈ End(R, +) be the map which is the sum of all possible words in σ, δ built with i letters σ and j − i letters δ. For example, f 0 0 = 1, f j j = σ i , f j 0 = δ j and f j j−1 = σ j−1 δ + σ j−2 δσ + · · · + δσ j−1 .…”

“…(2) We apply the proof of [21,Lemma 2.10]. Suppose that p(x) ∈ N (R[x; σ, δ]) for p(x) = a 0 + a 1 x + · · · + a n x n ∈ R[x; σ, δ].…”

Special Weak Properties of Generalized Power Series Rings

“…Reduced rings are symmetric by the results of Anderson and Camillo [1], but there are many nonreduced commutative (so symmetric) rings. As a generalization of symmetric rings, L. Ouyang introduced the notion of weak symmetric rings and showed that if R is an (α, δ)-compatible and reversible ring, then R is weak symmetric if and only if the Ore extension R[x; α, δ] is weak symmetric [16]. In the following, we investigate the weak symmetric property of the rings of generalized power series.…”

“…Any concept and notation not defined here can be founded in Ribenboim [17][18][19], Elliott and Ribenboim [6], and L. Ouyang [15][16].…”