Homomorphisms between two isomorphic rings ?

[robertjordan]

Homework Statement

True or False?
Let R and S be two isomorphic commutative rings (S=/={0}). Then any ring homomorphism from R to S is an isomorphism.

Homework Equations

R being a commutative ring means it's an abelian group under addition, and has the following additional properties:

i) a*(b+c)=a*b+a*c
ii) ab=ba
iii) a*(b*c)=(a*b)*c
iv) there exists an element e_R s.t. a*e_R=a for all a in R.



A "ring homomorphism" from R to S is a function f from R to S such that
i) f(a)*f(b)=f(a*b)
ii) f(a+b)=f(a)+f(b)
iii) f(e_S)=e_R

The Attempt at a Solution

BAck of the book says false

I thought to make f(a)=0_S for all a in S which would have worked as a counterexample but but it implies f(e_R)=0_S which by property (iii) of ring homomorphisms implies e_R=0_S which means a=a*e_S=a*0_S=0 so a=0 for all a in S but that means S={0} which is a contradiction.


Thanks for reading

 

[jbunniii]

The Attempt at a Solution

BAck of the book says false

I thought to make f(a)=0_S for all a in S which would have worked as a counterexample but but it implies f(e_R)=0_S which by property (iii) of ring homomorphisms implies e_R=0_S which means a=a*e_S=a*0_S=0 so a=0 for all a in S but that means S={0} which is a contradiction.

Consider for example a polynomial ring, such as $\mathbb{R}[x]$. Clearly it is isomorphic to itself. But there are many homomorphisms from $\mathbb{R}[x] \to \mathbb{R}[x]$ which are not isomorphisms. Can you find one?

 

[robertjordan]

Consider for example a polynomial ring, such as $\mathbb{R}[x]$. Clearly it is isomorphic to itself. But there are many homomorphisms from $\mathbb{R}[x] \to \mathbb{R}[x]$ which are not isomorphisms. Can you find one?

A homomorphism could me mapping every polynomial to its constant term (term without an x).
Say $P_{1}(x)=c_{n}x^n+...+c_{1}x+c_{0}$ and
$P_{2}(x)=k_{m}x^n+...+k_{1}x+k_{0}$

So $f(P_{1}(x)P_{2}(x))=c_{0}k_{0}=f(P_{1}(x)f(P_{2}(x))$
Also, $f(P_{1}(x)+P_{2}(x))=c_{0}+k_{0}=f(P_{1}(x)+f(P_{2}(x))$
And lastly, $f(1)=1$.

So indeed this is a ring homomorphism, but obviously it is not surjective or injective.Is this right?

 

[jbunniii]

A homomorphism could me mapping every polynomial to its constant term (term without an x).
Say $P_{1}(x)=c_{n}x^n+...+c_{1}x+c_{0}$ and
$P_{2}(x)=k_{m}x^n+...+k_{1}x+k_{0}$

So $f(P_{1}(x)P_{2}(x))=c_{0}k_{0}=f(P_{1}(x)f(P_{2}(x))$
Also, $f(P_{1}(x)+P_{2}(x))=c_{0}+k_{0}=f(P_{1}(x)+f(P_{2}(x))$
And lastly, $f(1)=1$.

So indeed this is a ring homomorphism, but obviously it is not surjective or injective.


Is this right?

Yes, that's the example I had in mind.