Bull. Korean Math. Soc. 2017 Vol. 54, No. 6, 1969-1980 https://doi.org/10.4134/BKMS.b160652 Published online July 3, 2017 Printed November 30, 2017

Gyu Whan Chang, Dong Yeol Oh Incheon National University, Chosun University

Abstract : Let $\Gamma$ be a nonzero torsionless commutative cancellative \linebreak monoid with quotient group $\langle \Gamma \rangle$, $R = \bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a graded integral domain graded by $\Gamma$ such that $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, $H$ be the set of nonzero homogeneous elements of $R$, $C(f)$ be the ideal of $R$ generated by the homogeneous components of $f \in R$, and $N(H) = \{f \in R \mid C(f)_v = R\}$. In this paper, we introduce the notion of graded $t$-almost Dedekind domains. We then show that $R$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain and $R_H$ is a $t$-almost Dedekind domains. We also show that if $R = D[\Gamma]$ is the monoid domain of $\Gamma$ over an integral domain $D$, then $R$ is a graded $t$-almost Dedekind domain if and only if $D$ and $\Gamma$ are $t$-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if $\langle \Gamma \rangle$ satisfies the ascending chain condition on its cyclic subgroups, then $R = D[\Gamma]$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain.