Thank you, everyone, who helped me last time for summer job extension question. I emailed my manager and she was positive with one-month extension. But in the interview, the total period for this summer job was until September which I will now complete. She has handed my contract that says I can stay until September. Also, I worked on some certain projects and proposed some future ideas to improve it. They really loved it. Now, do you think if it is appropriate to ask that they can offer me a job or I have some sort of future here in this company? I would want to work in this company and I am working enthusiastically on plans. Also, it's a big company and my manager is usually a very busy person. I do not know if they will give me a chance but they are very motivating and spare time to hear my plans and ask me to take actions.
Sure it is appropriate. Company's process resources to generate value. Your workforce is a resource. If they see a way to make use of it they will want it!
If you are new to the professional world it may take some time to get the confidence and realize that what you have to offer is actually a valuable asset and employers even compete about you.
Add to that, on a personal level it always feels good for you boss to know you enjoyed working there and want to continue doing so. He´ll remember you an even if they don´t have a position for you right now, they might in the future. Just keep checking until you get a definite no.