Can we write Allah's Name, Without "Al" before that Name? Example is writing "Qayyoom" instead of "Al-Qayyoom" This question arise because somebody has written only "Qayyoom" and mention that it is a Name of Allah.
Qayyoom is an adjective, it is not a private name. But Al-Qayyoom is a private adjective which refers only to Allah. Just like Ar-Rahman, Ar-Rahim, Al-Bari etc.
Ilah is the word in Arabic meaning god. You can claim that Zeus was a god, or Uzza. This won't be wrong literally. But Al-Ilah (Allah) means "one true god". It is like "The God", which is the one, everyone understands that you are not mentioning Zeus or Uzza.
Qayyoom is of the same type with merciful, but when we say All Merciful, we mean Allah.
Thus, you should put Al, if you want to refer to Allah.