In my abstract algebra text, the author uses "multiple" notation. Say you have a field $F$ that contains $a,b$. Consider some equation like $a^2 + 2ab + b^2 = 0$. The $2ab$ is meant to be shorthand for $ab + ab$ rather than the literal integer $2$ multiplied to $ab$.
In doing higher level computations in field theory, I encounter this notation and I'm always wondering whether or when I'm allowed to, say, divide both sides of $a^2+b^2 = -2ab$ by $2$. Can someone clarify the situations in which this multiple notation and integer multiplication coincide?