ON t-ALMOST DEDEKIND GRADED DOMAINS

Let ${\Gamma}$ be a nonzero torsionless commutative cancellative 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 RH 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 ${\langle}{\Gamma}{\rangle}$ isatisfies 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.