What is the ruling on using work computers to write research and save it on a flash drive without printing it out on paper?

The employee is entrusted with the work and the tools assigned to him, and he is not permitted to use his personal work tools except with explicit or customary permission. Ibn Taymiyyah holds that customary permission for permissibility, ownership, or disposition is like verbal permission, and both are established by whatever indicates them, whether by word or deed. One must refer to the employer's regulations to determine what is allowed and what is forbidden.