Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.

Show that T is the smallest subring of R containing X.

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- Mar 30th 2013, 07:37 PMBernhardSubring of R generated by X
Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.

Show that T is the smallest subring of R containing X. - Mar 31st 2013, 02:34 AMGusbobRe: Subring of R generated by X
This is actually a straightforward manipulation of definitions. You'll need to prove two things: is a subring, then is the smallest subring containing . Suppose your set of all subrings of containing is given by .

For the first one, you really just need to check that . The notation is a bit cumbersome, but this fact is actually quite trivial if you think about it. We also have by definition of .

The second bit is even easier: Let be a subring of containing . Since this is a subring of containing , we have by definition of . Using the basic property of intersections, this says that