Do properties of algebraic structures sometimes not carry over when their direct products are taken? I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are still rings (with identity), and it seemed really silly to go through all the steps in order to conclude what was extremely trivial. My question is, are there instances when this is not the case? Are there certain properties, or certain algebraic structures whereupon taking their direct products doesn't maintain the original algebraic structure/properties?

A direct product of integral domains is never an integral domain since $(1,0)\cdot(0,1)=0$. You can also consider PIDs; $\Bbb Z$ is a PID while $\Bbb Z\times\Bbb Z$ is no. The direct product of fields is not a field, say $\Bbb Q$ and $\Bbb Q\times\Bbb Q$.

There is also a problem of checking that if you don't use a canonical operation, checking that it still makes sense as an operation. For example, we can identify $\Bbb C$ with $\Bbb R^2$ by defining an operation $$(a,b)\cdot(c,d)=(ac-bd,ad+bc).$$ It is not immediately obvious that this is a well-defined operation that satisfies the axioms for a ring.