Complexes and Reals: The Impossibility of an Onto Ring Homomorphism

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In summary, an onto ring homomorphism is a surjective function between two rings that preserves the algebraic operations of addition and multiplication. It differs from a regular ring homomorphism in that it must be surjective. Some of its properties include preserving the identity, inverse, and zero elements. It can also be bijective, making it a one-to-one correspondence between the two rings. Onto ring homomorphisms are commonly used in abstract algebra and number theory, as well as in practical applications such as coding theory and cryptography.
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MathMike91
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Onto ring homomorphism C-->R

Why can't there be an onto ring homomorphism from the Complexes to the Reals?

The only property of the complexes that the rations don't have that I can think of is the guarantee of square roots- but I can't see how that would interfere with an onto function.
 
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Let f be an onto homomorphism. Let z be such that f(z)=-1. Consider a such that a²=z, then f(a)²=f(z)=-1. This can't happen...
 

FAQ: Complexes and Reals: The Impossibility of an Onto Ring Homomorphism

What is an onto ring homomorphism?

An onto ring homomorphism is a function between two rings, in this case from the ring C (complex numbers) to the ring R (real numbers), that preserves the algebraic operations of addition and multiplication. This means that the output of the function when operating on two elements of the first ring will be an element of the second ring.

How is an onto ring homomorphism different from a regular ring homomorphism?

An onto ring homomorphism is a type of ring homomorphism that is surjective, meaning that every element in the target ring has at least one pre-image in the source ring. This is not necessarily true for regular ring homomorphisms, which may only preserve the algebraic operations without being surjective.

What are some properties of an onto ring homomorphism?

An onto ring homomorphism has the following properties:

  • It preserves the identity element, meaning that the function maps the identity element of the source ring to the identity element of the target ring.
  • It preserves the inverse element, meaning that the function maps the inverse of an element in the source ring to the inverse of the same element in the target ring.
  • It preserves the zero element, meaning that the function maps the zero element of the source ring to the zero element of the target ring.

Can an onto ring homomorphism be bijective?

Yes, an onto ring homomorphism can be bijective, meaning that it is both surjective and injective. This means that every element in the target ring has exactly one pre-image in the source ring, making the function a one-to-one correspondence between the two rings.

How can onto ring homomorphisms be used in mathematical applications?

Onto ring homomorphisms are commonly used in abstract algebra and number theory. They can be used to study the structure and properties of different types of rings, and to prove theorems about the relationships between them. They also have practical applications in areas such as coding theory and cryptography.

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