Dear All, 


It is well-known that a group is finite iff the number of its subgroups is finite. 
Is there any similar result for rings and subrings ?? (Clearly we have to distinguish two cases: 


1. rings have identity and subrings have the same identity.
2. rings and subrings in general.



On the number of subrings 
This [article](http://dx.doi.org/10.1007/BF01350707) 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. 


