>>12394982if r is an element of a ring and n is a natural number, people do take the abuse of notation nr = r + r + ... + r. but you can't actually multiply n and r, since n is not in the ring.
similarly, we define r^n by r*r*...*r. this is not the same as if we defined an extra operation in the ring, ^, which for two elements r and s, gives an element r^s. we would need to provide properties of such a function.
again, you're confusing "binary operations on a ring" with "notation for repeated use of the binary operations on a ring on one element."
