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 β k , where p is a prime number and p ∤ k. We show that if the group of units of R has at most one subgroup of order p , then R ≅ A ⊕ B , where B is a finite ring of order k and A is a ring of cardinality p β which is one of the six explicitly described types. [ABSTRACT FROM AUTHOR]