While a virus is in the host chromosome,that virus is called a prophage. Temperate phage can remain as prophage indefinitely, sometimes decades, but can also, at any time jump out of the host chromosome and start lytic cycle where they left off, by beginning synthesis. Not all phage are temperate. Viruses with DNA genomes can be lysogenic. Don't match host genome so can't be incorporated into it.
All life forms except viruses have DNA genomes.
There is one special type of RNA virus that can be lysogenic. Overhead Let's see what's different about this process using animal cells as the host instead of bacteria. Remember animal cells have no cell wall, so easier penetration step for one thing. The envelope stays on the outside and only the nucleocapsid or sometimes just the viral genome goes into the cell. Other assembly is same as in phage.