MathDB

Let

R

be a commutative ring with

1

such that the number of elements of

R

is equal to

p^3

where

p

is a prime number. Prove that if the number of elements of

\text{zd}(R)

be in the form of

p^n

(

n \in \mathbb{N^*}

) where

\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}

, then

R

has exactly one maximal ideal.

3