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I've started to learn regular expressions in one of my classes

I know that a* is infinite, which leads me to believe that a*∅ is not infinite, since concatenating any language by a empty set is just empty set, similar to multiplying by zero. Is this line of thinking correct?

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    Yes, your argument is correct. Any set concatenated with the empty set gives the empty set. – 2017-02-16

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Yep, that's correct. An element of $a^*\emptyset$ would be a concatenation of an element of $a^*$ with an element of $\emptyset$. But there aren't any elements of $\emptyset$! So there is no way to build an element of $a^*\emptyset$, and it is empty.