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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.

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There is a third case, namely rings have to have identities but subrings don't necessarily share them. –  Qiaochu Yuan Dec 28 '12 at 9:07
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up vote 4 down vote accepted

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.

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The article seems to be referring to the second case. The result is clearly false in the first case, e.g. because $\mathbb{Z}$ has only one unital subring. –  Qiaochu Yuan Dec 28 '12 at 9:08
    
There is also a lot of interesting information here: books.google.ie/… –  Felix Goldberg Dec 28 '12 at 13:28
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