Abridged dependent clauses

Hi everyone!

I''ve been struggling to understand what an abridged statement is for, but haven't got it yet. Could you help me? I know that native speakers usually shorten certain phrases when the meaning might easily be implied. Thus, we have a reduction of the verb "to be" in utterances such as:

"while/As going to the supermarket, I remembered to take the shopping list with me!

After doing my homework, I'll go to play soccer with my friend.

Acording to what I taught myself, to make this reducation, it's necessary that the subject of both clauses, the dependent and the main clause, is the same. As for me, It wouldn't make sense , if there are two different subjects within the same abriged sentence. The utterance implies, the goer remembered to take the shopping list on his way to the supermarket. (during the time he was walking to the super) is this ok?

Grammatically speaking, the reduction is just a way of reducing conjunctions. Is that easy?