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10

Expert-verifiedFound in: Page 80

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**If ${\mathit{R}}$ is a ring with identity, and ${\mathit{f}}{\mathbf{:}}{\mathit{R}}{\mathbf{\to}}{\mathit{S}}$ is a homomorphism from ${\mathit{R}}$ to a ring ${\mathit{S}}$ , prove that ${\mathit{f}}{\left({1}_{R}\right)}$ is an idempotent in ${\mathit{S}}$ . [Idempotent were defined in Exercise 3 of Section 3.2.]**** **

Hence it is proved that $f\left({1}_{R}\right)$ is an idempotent in $S$ .

If any ring $R$ has elements such that, $a,b\in R$, then, addition and multiplication of the function of its elements is respectively given by:

$f\left(a+b\right)=f\left(a\right)+f\left(b\right)\phantom{\rule{0ex}{0ex}}f\left(ab\right)=f\left(a\right)f\left(b\right)$

We have a homomorphism of rings as$f:R\to S$:

In this case, we get:

$f\left({1}_{R}\right)=f\left({1}_{R}{1}_{R}\right)\phantom{\rule{0ex}{0ex}}=f\left({1}_{R}\right)f\left({1}_{R}\right)\phantom{\rule{0ex}{0ex}}=f{\left({1}_{R}\right)}^{2}$

Hence proved, $f\left({1}_{R}\right)$ is an idempotent in $S$ .

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