The Singly Generated Unital Rings with Only Finitely Many Unital Subrings

Abstract

The rings of the title are characterized. In view of earlier work on this problem, the main contribution here is the following result. Let R be a (commutative unital) ring extension of ℤ of the form ℤ[t] which is not an integral domain and which is not integral over ℤ. Then R has only finitely many (unital) subrings if and only if there exist nonzero integers a, b with at = b such that the minimal positive such a and the corresponding b satisfy (i) is integral over ℤ and (ii) there does not exist a prime number p such that ker(ϕ)⊆ p ℤ[X], where ϕ is the (unital) ring homomorphism ℤ[X] → R sending X to t. It is also proved that if T is any finite ring and n any nonzero integer, then T × ℤ[1/n] has only finitely many subrings.