The article explores relationships between different types of mathematical properties in graded integral domains. By analyzing the ideal generated by the components of elements in the domain, the researchers found that certain properties of the domain are linked to properties of the ideal generated by polynomials in the domain. Specifically, they discovered that a domain is a graded Prüfer-$\star$-multiplication domain if and only if a certain related domain is a Prüfer domain, which in turn is equivalent to another domain being a Bézout domain. Additionally, the researchers determined conditions under which a specific type of domain is a principal ideal domain.