Muhly local domains and Zariski's theory of complete ideals

Let $(R,\M)$ be a two-dimensional Muhly local domain, that is, a two-dimensional integrally closed Noetherian local domain with algebraically closed residue field and with the associated graded ring $\text{gr}_{\M}R$ an integrally closed domain. In this paper we show that a number of fundamental results of Zariski's theory of complete ideals in two-dimensional regular local rings are not necessarily valid in $R$. However, if the associated graded ring $\text{gr}_{\M}R$ satisfies an additional assumption as in work of Muhly and Sakuma, then we are able to show that ``any product of contracted ideals is contracted'' holds in $R$ if and only if $R$ has minimal multiplicity.