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In my lecture today, the lecturer mentioned in passing that there are unusual results when you look at combination of elements of a ring, and whether they form a subring or not.
More specifically he said that if a and b are subrings of a ring, this implies that a-2b, 2a-b and 3a-b are subrings, while a-3b is not. He didn't go into any detail why this is the case (perhaps we haven't learnt enough to explain it yet) but I was wondering whether anyone could shed any light on why this is the case?

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Can you explain your notation here? –  Qiaochu Yuan Dec 17 '10 at 8:36
I will try although I think this may be confusing me as well!! a and b are subrings of a ring, say A where a-b is the elements in a but not in b, and b-a is the elements in b but not a. –  Lucy Marshall Dec 18 '10 at 14:58
If you take $a=b=\mathbb{Z}$ then $a-2b$ is just the odd integers, which is not a ring. More generally, if a,b are subrings, then $a-b$ doesn't contain the zero element, so it cannot be a ring. –  Prometheus Dec 18 '10 at 17:39

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