A higher-dimensional generalization of almost perfect domains

An integral domain is called almost perfect if all its proper quotients are perfect rings; such domains are necessarily one-dimensional. This paper studies a higher-dimensional generalization of almost perfect domains via the [Formula: see text]-operation. We define a [Formula: see text]-almost perfect domain as a domain [Formula: see text] for which the [Formula: see text]-factor ring [Formula: see text] of [Formula: see text] is a perfect ring for every nonzero [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. The class of [Formula: see text]-almost perfect domains properly contains almost perfect domains, Krull domains, and strong Mori domains of [Formula: see text]-dimension one. Several characterizations of [Formula: see text]-almost perfect domains are established. In particular, it is proved that a domain [Formula: see text] is [Formula: see text]-almost perfect if and only if it is of finite [Formula: see text]-character and [Formula: see text]-locally almost perfect. A module-theoretic characterization of [Formula: see text]-almost perfect domains is given by using the notion of [Formula: see text]-semi-artinian modules: a domain [Formula: see text] is [Formula: see text]-almost perfect precisely when its quotient module [Formula: see text] is [Formula: see text]-semi-artinian and [Formula: see text] is [Formula: see text]-local, where [Formula: see text] is the quotient field of [Formula: see text] and [Formula: see text] is the hereditary torsion theory induced by the [Formula: see text]-operation. Furthermore, in order to provide some original examples, we also characterize [Formula: see text]-almost perfect domains in certain types of pullback constructions. Finally, we show that a domain [Formula: see text] is a Krull domain if and only if [Formula: see text] is a Prüfer [Formula: see text]-multiplication domain which is [Formula: see text]-almost perfect, and that every [Formula: see text]-almost perfect domain is a [Formula: see text]-Matlis domain.