It depends on how you define rings. Some authors insist that a ring must contain a nonzero multiplicative identity. Others do not insist on this condition. If you use the second alternative, then it is possible for a ring to contain just one element, namely the zero element.
A nonzero ring would then be a ring that contains at least one nonzero element.
Here’s a hint to get you started. Let be a nonzero element in and be a nonzero element in Then and are nonzero elements in Now multiply them together. What do you get?