Let R be a ring such that 0 is both the additive identity an

Let R be a ring such that 0 is both the additive identity and the multiplicative identity. Show that there is only one element in R.

Solution

Let a be an arbitrary element of the ring R. Since 0 is the multiplicative identity, we have 0 *a = a *0 = a. However, 0 * a = a *0 = 0 for all a in R. Therefore a = 0. Thus, there is only one element in R i.e. 0.