Final answer:
To prove that a ring r is the zero ring if 0 = 1, you consider any element a in r, show that a = a * 1, substitute 1 with 0, and use the fact that a * 0 = 0, concluding a must equal 0, so r must be {0}.
Step-by-step explanation:
You're asking how to show that if 0 = 1 in a ring r, then the ring is equal to {0}, which is the zero ring. To prove this, let's consider any element a in the ring r. By the definition of a ring, we can multiply any element by the multiplicative identity (1 in most rings) to get the element itself: a * 1 = a. But since 0 = 1 in this ring, we can substitute 0 for 1, giving us a * 0 = a. However, anything multiplied by 0 in a ring gives 0, thus a * 0 = 0. This would mean a = 0 for any a in r, hence the only element in r is 0 and thereby r is the zero ring {0}.