Does the "necessarily" part in the question make a difference?

Question

"Given an associative F-algebra do there necessarily exist elements a1.a2∈K satisfying a1∗a2=a?"

I have a question to add to the one linked below:

Given an associative $F$-algebra do there necessarily exist elements $a_1,a_2 \in K$ satisfying $a_1*a_2=a$?

My question: Will there always exist a1,a2∈K satisfying a1∗a2=a? I thought for F-algebra "*" didn't necessarily have an identity element, or associativity. I know it tells us we have an associative F-algebra, but why do those elements 'have' to exist? There were 2 instances mentioned where they do exist, but I am wondering if they 'have to exist.'

I may just be paying too much attention to one word, but I want to make sure I understand the subject. Thanks!