Finite unitary rings with a single subgroup of prime order of the group of units

Let R be a unitary ring of finite cardinality P^k, where p is a prime number and $p\nmid k$. We show that if the group of units of $R$ has at most one subgroup of order $p$, then $R\cong A⨁B,$ where $B$ is a finite ring of order $k$ and $A$ is a ring of cardinality $p^{\beta }$ which is one of the six explicitly described types.