Noetherian properties in composite generalized power series rings

Abstract

Let (Γ,≤)({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let Γ⁎=Γ\{0}{{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\}. Let D⊆ED\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. SetD+〚EΓ⁎,≤〛≔f∈〚EΓ,≤〛|f(0)∈DandD+〚IΓ⁎,≤〛≔f∈〚DΓ,≤〛|f(α)∈I,forallα∈Γ⁎.\begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array}In this paper, we give necessary conditions for the rings D+〚EΓ⁎,≤〛D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when (Γ,≤)({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+〚EΓ⁎,≤〛D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when (Γ,≤)({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+〚IΓ⁎,≤〛D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when (Γ,≤)({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+(X1,…,Xn)E[X1,…,Xn]D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+(X1,…,Xn)I[X1,…,Xn]D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

Keywords

• d+〚eγ⁎,≤〛,d+〚iγ⁎,≤〛
• generalized power series ring
• noetherian ring
• 13a02
• 13a15
• 13b35
• 13e05