Ring Homomorphism: Show Phi(1) Is Unity for R

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In summary, the conversation discusses a proof that if R and R' are rings, and phi: R to R' is a ring homomorphism where phi[R] is not identically 0', then if R has unity 1 and R' has no 0 divisors, then phi(1) is a unity for R'. The conversation also explores the definition of a subring and its properties, and provides a proof for the fact that 1 must be a unity for R given the conditions stated.
  • #1
ehrenfest
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Homework Statement


Let R and R' are rings and phi: R to R' is a ring homomorphism such that phi[R] is not identically 0'. Show that if R has unity 1 and R' has no 0 divisors, then phi(1) is a unity for R'.

Homework Equations


The Attempt at a Solution



Its relatively obviously why phi(1) has to be unity for the subring phi[R]. I don't see why phi(1)r' has to be r' for every r' in R'.
 
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  • #2
CAN subring have an identity other than the ring identity?
 
  • #3
I don't know.

If S is a subring of R and 1 is a unity of S, then is it true that 1r=r for an r in R? I don't see why?

How do I use the fact that R' has no 0 divisors.
 
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  • #4
Recall that ehrenfest is not using the usual notion of a 'ring'; he uses a variant that omits '1' from the definition. So if the ring and subring have '1', his definition of subring doesn't require them to be the same.

There is a simple example of a subring with a different unit: consider the ring of 2x2 matrices, and the subring of those whose upper-left entry is the only nonzero.
 
  • #5
And in that example, there are zero divisors. Use the fact that lack of zero divisors admits a cancellation rule, i.e., r's' = r't' => s' = t'.

At least, I think that's right.
 
  • #6
A ring <R,+,*> is a set R together with two binary operations + and * such that the following axioms are satisfied:
1) <R,+> is an abelian group
2) Multiplication is associative
3) For all a,b,c in R, a*(b+c)=(b+c)*a=a*b+a*c

A subring is a subset of R that is also a ring under + and *.

Now, how to solve the problem...

EDIT: sorry I didn't read the post above this when I wrote that
 
  • #7
If nothing springs to mind, then just try exploring. e.g. what properties does the unit of a subring have?
 
  • #8
Here is the proof that if R is a ring with no 0 divisors and S is a nonzero subring of R with unity 1, then 1 MUST be a unity of R:

11=1 implies that r11=r1 implies that r1=r by cancellation

11=1 implies that 11r=1r implies that 1r=r by cancellation

Is that right?
 

Related to Ring Homomorphism: Show Phi(1) Is Unity for R

1. What is a ring homomorphism?

A ring homomorphism is a function between two rings that preserves the ring structure. This means that the function respects the addition and multiplication operations of the rings.

2. What does it mean to show that phi(1) is unity for R?

Showing that phi(1) is unity for R means proving that the image of the multiplicative identity element of one ring is the multiplicative identity element of the other ring under the ring homomorphism.

3. Why is it important to show that phi(1) is unity for R?

It is important to show that phi(1) is unity for R because it is a fundamental property of ring homomorphisms. It ensures that the ring homomorphism is preserving the identity element, which is crucial in maintaining the structure and properties of the rings involved.

4. How can we prove that phi(1) is unity for R?

The proof involves showing that the image of the multiplicative identity element of one ring, say R, under the ring homomorphism phi is equal to the multiplicative identity element of the other ring, say S. This can be done by applying the definition of a ring homomorphism and using the properties of the identity element in both rings.

5. Are there any other properties of ring homomorphisms that are important to know?

Yes, there are several other important properties of ring homomorphisms, such as preserving inverses, zero divisors, and units. These properties help in understanding the behavior of ring homomorphisms and their impact on the rings involved.

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