Does the equality $x^n-y^n =(x-y)\sum_{i=0}^{n-1} x^i y^{n-1-i}$ hods for a commutative ring?

Question

Given a commutative ring $(R,+,*)$ I would like to understand if the equality $$ \tag{1}\label{1}x^n-y^n =(x-y)\sum_{i=0}^{n-1} x^i y^{n-1-i} $$ holds for any $x,y\in R$. In fact, here I found a proof of the above identity where $x$ and $y$ are reals but actually the proof is by induction using only the usual properties of any commutative ring so that I believe \eqref{1} holds for any commutative ring: however, into the comments the professor Brandemburg seems state that induction does not work so that I thought to put here an answer where I ask if \eqref{1} holds for commutative ring and so if it is possibile to prove it with induction. So clould someone help me, please?