SPECIAL WEAK PROPERTIES OF GENERALIZED POWER SERIES RINGS

Title & Authors
SPECIAL WEAK PROPERTIES OF GENERALIZED POWER SERIES RINGS
Ouyang, Lunqun;

Abstract
Let $\small{R}$ be a ring and $\small{nil(R)}$ the set of all nilpotent elements of $\small{R}$. For a subset $\small{X}$ of a ring $\small{R}$, we define $\small{N_R(X)=\{a{\in}R{\mid}xa{\in}nil(R)}$ for all $\small{x{\in}X}$}, which is called a weak annihilator of $\small{X}$ in $\small{R}$. $\small{A}$ ring $\small{R}$ is called weak zip provided that for any subset $\small{X}$ of $\small{R}$, if $\small{N_R(Y){\subseteq}nil(R)}$, then there exists a finite subset $\small{Y{\subseteq}X}$ such that $\small{N_R(Y){\subseteq}nil(R)}$, and a ring $\small{R}$ is called weak symmetric if $\small{abc{\in}nil(R){\Rightarrow}acb{\in}nil(R)}$ for all a, b, $\small{c{\in}R}$. It is shown that a generalized power series ring $\small{[[R^{S,{\leq}}]]}$ is weak zip (resp. weak symmetric) if and only if $\small{R}$ is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring $\small{[[R^{S,{\leq}}]]}$ in terms of all weak associated primes of $\small{R}$ in a very straightforward way.
Keywords
weak annihilator;weak associated prime;generalized power series;
Language
English
Cited by
1.
McCoy property and nilpotent elements of skew generalized power series rings, Journal of Algebra and Its Applications, 2016, 1750183
2.
Nilpotent elements and nil-Armendariz property of skew generalized power series rings, Asian-European Journal of Mathematics, 2016, 1750034
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