- Interior design is an interesting thing to study.
- Studying English takes a lot of time.
- History can be cruel.
- They said "Money is power".
Therefore, this sentence is correct:
Some people do not think interior design is an important part of life.
It falls under the category of an "abstract noun" for an idea, quality or state, in this case, subject matter.
These types of nouns (the ones above) when used as such (as a topic being discussed or commented on) do not require a determiner.
- a part of life = a generality, no plural.
a part of their lives=specifically referring to some group.
Some people do not think interior design is an important part of life.
[a generality about life; life is an abstract noun here]
Some people do not think interior design is an important part of their lives. [specifically referring to some people and their actual lives, not a generality and made specific by the pronoun their; lives is not abstract here.]
[Please note: I have only dealt with the lack of a determiner for an abstract noun. I am not dealing with every single instance of determiners such as "The interior design of this house", etc. In my experience, it's best to try and grasp one major point at a time.]