The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring

Abstract

In this paper, we give conditions on a unital non-associative ring $R$ and additive map $σ: R\longrightarrow R$ so that the non-associative skew power series ring, denoted by $R[[X;σ]]$ and the non-associative skew Laurent series ring denoted by $R((X;σ))$ are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of $R[[X;σ]]$ and $R((X;σ))$, respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for $R[[X;σ]]$ and $R((X;σ))$, we introduce and utilize the right (resp. left) quasi-associative property of $R$, which is equivalent to $rR$ (resp. $Rc$) being one-sided ideals for all $r\in R$ (resp. $c\in R$).