Why do we have to do the same things to both sides of an equation? Forgive me in advanced if this is a trivial question.
This convention makes perfect sense to me intuitively, but is there any rigorous underpinning to it?
I'm beginning to read through an abstract algebra textbook, and soon after establishing what a ring is, it all of the sudden added to both sides of the equation to prove a(0)=(0)a=0. It seemed a little premature to me. 

Here's a short example of doing the same thing to both sides of an equation, and a justification for each step, using only substitution and the field axioms. The magic happens in the first two steps.
Statement:
\begin{align*}
a+c=b+c \Rightarrow a=b
\end{align*}
Proof:
\begin{align*}
(a+c)+(-c)&=(a+c)+(-c) \tag{property of =} \\
(a+c)+(-c)&=(b+c)+(-c) \tag{substitution}\\
a+(c+(-c))&=b+(c+(-c)) \tag{associativity}\\
a+0&=b+0 \tag{inverses}\\
a&=b \tag{identity}
\end{align*}
$\square$
