-eum and -ium?

[David / Amicus]

Do I understand correctly?

A word ending in "eum" means a building, temple etc. like Trajaneum?

But "ium" would be like a school, institution etc.. Like Trajanium could
be a school named in honor of Trajan?

[David Rodericius]

I don't think the Romans made such a strict distinction between the
two. Words ending either in "-eum" or "-ium" are neuter nouns derived
from adjectives "-eus, -ea, -eum" and "-ius, -ia-, -ium",
respectively. The adjectives themselves simply mean 'of or pertaining
to X'. There are many adjectives formed from the name of Nero, for
instance: Neroneus/a/um, Neronianus/a/um, Neronius/a/um. There would
have always been an implied noun, which in the case of "Trojaneum"
could be "templum"; however, the implied noun could just as well be
"collegium". In Rome, many theological colleges are commonly known
this way, e.g., the Russicum, the Gregorianum, the Angelicum, the
implied noun being "collegium".

Rod.