Abstract
AbstractLet(Γ,≤)({\mathrm{\Gamma}},\le )be a strictly ordered monoid, and letΓ⁎=Γ\{0}{{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\}. LetD⊆ED\subseteq Ebe an extension of commutative rings with identity, and letIbe a nonzero proper ideal ofD. 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 ringsD+〚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 ringsD+〚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 ringD+〚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 ringsD+(X1,…,Xn)E[X1,…,Xn]D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}]andD+(X1,…,Xn)I[X1,…,Xn]D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}]to be Noetherian.
Highlights
Throughout this paper, a monoid means a commutative semigroup with identity element
A monoid Γ is said to be cancellative if every element in Γ is cancellative, i.e., for every α, β, γ ∈ Γ, α + β = α + γ implies β = γ
We denote by G(Γ) the largest subgroup of Γ, i.e., G(Γ) ≔ {α ∈ Γ | α + β = 0, for some β ∈ Γ}
Summary
Throughout this paper, a monoid means a commutative semigroup with identity element. The operation is written additively and the identity element is denoted by 0, unless otherwise stated.A monoid Γ is said to be cancellative if every element in Γ is cancellative, i.e., for every α, β, γ ∈ Γ, α + β = α + γ implies β = γ. Let D ⊆ E be an extension of commutative rings with identity and (Γ, ≤) a positively ordered monoid. (cf [2, 5.2]) Let D be a commutative ring with identity and (Γ, ≤) a positively ordered monoid.
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