### 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 ⊆ E D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D . Set D + 〚 E Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 E Γ , ≤ 〛 | f ( 0 ) ∈ D and D + 〚 I Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 D Γ , ≤ 〛 | f ( α ) ∈ I , for all α ∈ 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 + 〚 E Γ ⁎ , ≤ 〛 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 + 〚 E Γ ⁎ , ≤ 〛 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 + 〚 I Γ ⁎ , ≤ 〛 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 + ( X 1 , … , X n ) E [ X 1 , … , X n ] D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D + ( X 1 , … , X n ) I [ X 1 , … , X n ] D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.