TY - JOUR
T1 - Noetherian properties in composite generalized power series rings
AU - Lim, Jung Wook
AU - Oh, Dong Yeol
N1 - Publisher Copyright:
© 2020 Jung Wook Lim and Dong Yeol Oh, published by De Gruyter 2020.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - Let (Γ, ≤) ({\mathrm{\Gamma}},\le) be a strictly ordered monoid, and let Γ âŽ = 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 + ãš E Γ âŽ, ≤ ã:= f â ãš E Γ, ≤ ã | f (0) â D and D + ãš I Γ âŽ, ≤ ã:= f â ãš D Γ, ≤ ã | f (α) â I, for all α â Γ âŽ. \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 + ãš E Γ âŽ, ≤ ã 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 + ãš E Γ âŽ, ≤ ã 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 + ãš I Γ âŽ, ≤ ã 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.
AB - Let (Γ, ≤) ({\mathrm{\Gamma}},\le) be a strictly ordered monoid, and let Γ âŽ = 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 + ãš E Γ âŽ, ≤ ã:= f â ãš E Γ, ≤ ã | f (0) â D and D + ãš I Γ âŽ, ≤ ã:= f â ãš D Γ, ≤ ã | f (α) â I, for all α â Γ âŽ. \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 + ãš E Γ âŽ, ≤ ã 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 + ãš E Γ âŽ, ≤ ã 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 + ãš I Γ âŽ, ≤ ã 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.
KW - D+ãšEã,D+ãšIã
KW - generalized power series ring
KW - Noetherian ring
UR - http://www.scopus.com/inward/record.url?scp=85099335723&partnerID=8YFLogxK
U2 - 10.1515/math-2020-0103
DO - 10.1515/math-2020-0103
M3 - Article
AN - SCOPUS:85099335723
SN - 1895-1074
VL - 18
SP - 1540
EP - 1551
JO - Open Mathematics
JF - Open Mathematics
IS - 1
ER -