This article says that an associative ring with a finite number of subrings is finite.

As pointed out below, the result does not hold for rings with a unit as witnessed by $ \mathbb{Z}$ which has no proper subrings. Also, $ \mathbb{Z}[1/2]$ has two subrings including itself and $ \mathbb{Z}[1/6]$ has four. This shows that there is an infinite ring with $2^k$ subrings. What other examples are there which are commutative integral domains? I don't immediately see any. I think this is a ring with $3$ proper subrings: $ \mathbb{Z}[x]$ where $x^3=0\ne x^2$ and $7x=0.$ Of course $7$ can be replaced by any other prime.