These are relatively simple questions, but I can't seem to find anything on tensor products and p-adic integers/numbers anywhere, so I thought I'd ask.

My first question is: given some $\mathbb{Z}_p$, $\mathbb{Q}_p$ can be constructed as its field of fractions. Is the tensor product $\mathbb{Z}_p \otimes \mathbb{Q} $ also equal to $\mathbb{Q}_p$?

My second question is: given some *composite* p, you can still construct a ring (now with zero divisors) that can be constructed as the inverse limit of $\mathbb{Z}/p^n\mathbb{Z}$. Any such p has a finite prime factorization, so is there a way to construct this non-integral-domain ring of composite p-adics by taking some sort of product of the rings of p-adics of its prime factors? For instance, can you construct the 10-adics by taking the direct product of the 2-adics and the 5-adics, or perhaps is it the tensor product, or...?

My last question is: I haven't seen much about taking tensor products about p-adic integers in general; I've only seen stuff about taking direct products, as in the case of the profinite completion $\hat{\mathbb{Z}}$ of the integers. However, I find the tensor product to be of particular interest, since it's a coproduct in the category of commutative rings. So what, in general, do you get if you take the tensor product of two rings of p-adic integers? And I'm especially curious to know, what do you get if you take the tensor product of all of the rings of p-adic integers, rather than the direct product in the case of $\hat{\mathbb{Z}}$?