How is it that a commutative ring F(R) defined by the operations of pointwise addition and pointwise multiplication contains elements f != 0, 1 that satisfy f^2 = f?

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- November 22nd 2009, 06:21 PMjohnt4335Commutative Ring
How is it that a commutative ring F(R) defined by the operations of pointwise addition and pointwise multiplication contains elements f != 0, 1 that satisfy f^2 = f?

- November 24th 2009, 06:51 AMux0
- November 24th 2009, 08:08 AMclic-clac
Hi

If by F(R) you mean the set of all maps from to where is a ring, then each element of can be seen as a "sequence" where , and the binary operations you defined over it to obtain a ring are:

addition: ,

multiplication: ,

with as zero and identity element which are obviously idempotent.

When has more than two elements, can you find very simple elements of other than and (but similar...) that are idempotent? - November 24th 2009, 09:40 AMjohnt4335
Ohh okay, so you mean something along the lines of the matrix I.

- November 24th 2009, 09:54 AMclic-clac
I guess; what I mean is any such that is idempotent.

So if is finite and has elements, there are*at least*elements in that are idempotent.