an anilox printing unit includes , as inking unit rolls , an ink applicator roll and an engraved roll mounted in rapid change roll sockets , for removing the engraved roll from the roll sockets and inserting another engraved roll into the roll sockets by an operator . the engraved roll is hollow and a temperature control fluid flows therethrough . the engraved roll and the ink applicator roll each have bearer rings . a device which presses the bearer rings of one inking unit roll against the bearer rings of the other inking unit roll has springs for compensating for diameter differences as a result of manufacturing tolerances between the bearer rings of the engraved roll and the bearer rings of the other engraved roll and for compensating for thermally induced diameter changes of the bearer rings of the engraved roll . a printing press having an anilox printing unit is also provided .