Homomorphisms between two isomorphic rings ?

In summary: It satisfies all the properties of a ring homomorphism, but it is not an isomorphism since it is not injective. Therefore, the statement is false.
  • #1
robertjordan
71
0

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 eR s.t. a*eR=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(eS)=eR

The Attempt at a Solution


BAck of the book says false

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


Thanks for reading
 
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  • #2
robertjordan said:

The Attempt at a Solution


BAck of the book says false

I thought to make f(a)=0S for all a in S which would have worked as a counterexample but but it implies f(eR)=0S which by property (iii) of ring homomorphisms implies eR=0S which means a=a*eS=a*0S=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?
 
  • #3
jbunniii said:
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?
 
  • #4
robertjordan said:
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.
 

Related to Homomorphisms between two isomorphic rings ?

1. What is a homomorphism between two isomorphic rings?

A homomorphism between two isomorphic rings is a function that preserves the algebraic structure of the rings. It maps elements in one ring to corresponding elements in the other ring while preserving the addition and multiplication operations.

2. How do you prove that two rings are isomorphic?

To prove that two rings are isomorphic, you need to show that there exists a bijective homomorphism between the two rings. This means that the function must be both one-to-one and onto, and it must also preserve the ring operations of addition and multiplication.

3. What is the significance of homomorphisms between isomorphic rings?

Homomorphisms between isomorphic rings play a crucial role in understanding the properties and structure of rings. They allow us to identify and relate elements in one ring to elements in another ring, making it easier to study and analyze their properties and relationships.

4. Can two rings be isomorphic but not have any homomorphisms between them?

No, two rings cannot be isomorphic without having any homomorphisms between them. Isomorphism implies the existence of a bijective homomorphism between two rings, so if there are no homomorphisms between them, they cannot be isomorphic.

5. Is there a limit to the number of homomorphisms between two isomorphic rings?

Yes, there is a limit to the number of homomorphisms between two isomorphic rings. If the two rings are finite, then the number of homomorphisms between them is also finite. However, if the rings are infinite, there can be an infinite number of homomorphisms between them.

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