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Careful when you talk about mathematical operations, because they don't need to always be reversible, or "invertible". But in this case you got it right, under the <em>field</em> of real numbers, multiplication is invertible <b>if</b> you exclude 0, for the reasons you already stated.
Note however that sometimes mathematicians will say things like 1/0 = infinity, but this is really just shorthand for the limit. I think a more interesting question is the value of 0^0. Calculus treats this as undefined. Not everyone agrees though. Some discrete mathematicians and logicians consider 0^0 = 1, and not just formally to make sums look pretty, but meaningfully. Fun topic.
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