Suppose that Φ: R ⇒ S is a ring homomorphism and x belongs to R is idempotent. Show Φ(x) is idempotent in S.

Favorites|Homepage
Subscriptions | sitemap

HOME > Mathematics > Suppose that Φ: R ⇒ S is a ring homomorphism and x belongs to R is idempotent. Show Φ(x) is idempotent in S.

Suppose that Φ: R ⇒ S is a ring homomorphism and x belongs to R is idempotent. Show Φ(x) is idempotent in S.

[From: ] [author: ] [Date: 11-04-22] [Hit: ]

....= Φ(x) Φ(x),.......

I know that we have to show (Φ(x))^2 = Φ(x) and that x is idempotent if x^2 = x, but I don't know where to go after that

-

Given an idempotent x:

Φ(x) = Φ(x^2), since x = x^2
.......= Φ(x) Φ(x), since Φ is a ring homomorphism
.......= [Φ(x)]^2.

I hope this helps!

1

keywords: and,homomorphism,that,Phi,belongs,is,ring,Suppose,in,idempotent,to,rArr,Show,Suppose that Φ: R ⇒ S is a ring homomorphism and x belongs to R is idempotent. Show Φ(x) is idempotent in S.

New

Hot