Let S be the set of all scalar homomorphisms of an algebra X. Then the radical rad{X} of X is defined to be rad{X}=intersection_{f in {S}} {Ker{f}}.

Can anyone help me prove that a radical of an algbera is an ideal?

Thanks

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- July 27th 2012, 02:29 AMCairoRadical is an Ideal
Let S be the set of all scalar homomorphisms of an algebra X. Then the radical rad{X} of X is defined to be rad{X}=intersection_{f in {S}} {Ker{f}}.

Can anyone help me prove that a radical of an algbera is an ideal?

Thanks - July 27th 2012, 05:11 AMgirdavRe: Radical is an Ideal
Take in the radical , an element of the algebra and a scalar homomorphism. Show that .

- July 28th 2012, 09:54 AMCairoRe: Radical is an Ideal
- July 28th 2012, 09:58 AMgirdavRe: Radical is an Ideal
What's wrong with this?

- July 28th 2012, 10:09 AMCairoRe: Radical is an Ideal
Do we not need to prove that it is a subspace of X too?

I'm really confused and having be drowning in functional analysis problems all day. Are you able to offer a full proof? - July 28th 2012, 10:14 AMgirdavRe: Radical is an Ideal
Yes, we also need to check that the radical is stable by sums: the sum of two elements of the ideal still is in the ideal. After that, I think we get all the condition that an ideal needs to satisfy (ideal in sense of ring theory, I mean).

- July 28th 2012, 10:18 AMCairoRe: Radical is an Ideal
So if x, y in Ker(f) and lambda, mu in C, then

f(lambda(x) + mu(y)) = f(lambda(x)) + f(mu(y))= lambda f(x) + mu f(y) = lambda.0 + mu.0 = 0, so that lambda(x)+mu(y) in Ker(f)?