Characteristic of an Integral Domain

Let $R$ be an integral domain. Suppose there exists a nonzero $y$ such that $y + y + \ldots + y$ (n times) = 0, where $n$ is an integer greater than 1. Show that $x + x + \ldots + x$ (n times) = 0 for all $x$ in $R$.

In this problem, we explore a fundamental property related to the characteristic of an integral domain. Integral domains are an important class of rings which, among other things, possess no zero divisors. A key inquiry in ring theory is identifying the characteristic of a ring, which is the smallest positive number n such that adding the multiplicative identity to itself n times yields zero. If no such number exists, the ring is said to have characteristic zero. The problem gives us a crucial condition: there exists a non-zero element y of the integral domain such that the sum of n copies of y is zero for some integer n greater than one. This explicitly implies that the integral domain cannot have characteristic zero, since a non-zero element repeated n times equates to zero. Through this, we demonstrate that this sum property must hold for all elements x in the integral domain, leading us to conclude that it has characteristic n. Understanding such properties not only sheds light on the behavior of abstract algebraic structures but also aids in grasping how different algebraic systems generalize or restrict various arithmetic laws. Often, problems like these guide us in appreciating how integral domains contribute foundational insights into ring theory and abstract algebra as a whole.