A "$v$-operation free" approach to Pr\"ufer $v$-multiplication domains

The so called Pr\"ufer $v$-multiplication domains (P$v$MD's) are usually defined as domains whose finitely generated nonzero ideals are $t$-invertible. These domains generalize Pr\"ufer domains and Krull domains. The P$v$MD's are relatively obscure compared to their very well known special cases. One of the reasons could be that the study of P$v$MD's uses the jargon of star operations, such as the $v$-operation and the $t$-operation. In this paper, we provide characterizations of and basic results on P$v$MD's and related notions without star operations.