From what I understand (and I'm still very much a beginner), 'ohayou' is the informal way to say 'good morning' and it basically just means 'morning'. You might just say 'ohayou' to a friend or coworker. 'Ohayou gozaimasu' is a more polite, more formal way to say 'good morning' as you might to a stranger or your boss.
I was once working in a hotel full of japanese guests, years ago, and as I said "Ohayou" they laughed and told me I had to say "gozaimasu". Then I started to pay attention to the way they said it amongst each other and they basically said "gozaimasu" leaving the "Ohayou" out, as a compliment at breakfast. And they were coworkers. So I'd say unless the other person is your friend, like real friend, most important part is "gozaimasu".
"Ohayou" (casual/informal) and "ohayou gozaimasu" (polite/formal/keigo) both mean "good morning in english. Question should be changed.
(1) Question: "Good morning! (polite)"
Answer: "Ohayou gozaimasu"
(2) Question: "Good morning! (casual)"
If you teach both casual and polite, you should also ask both. And most importantly, specify which one you're asking.
ございます is an old form, which means more or less the same as あります from the verb ある (to have/there is/are) in modern Japanese. As such, it's only used on its own when you try to speak in an old-fashioned way (samurai drama etc.) or if you use 敬語（けいご）. Aside from that, it's not really used on its own (unless I forgot something). It cannot mean "good morning" without おはよう.