Homomorphism of Rings & Is Map f:C->Z a Homo?

MHB
Thread starter Stephen88
Start date Mar 30, 2012
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Rings

In summary, a homomorphism of rings is a map between two rings that preserves the ring structure, including addition, multiplication, and identity elements. It differs from an isomorphism in that it does not necessarily have an inverse or one-to-one correspondence. Important properties of a homomorphism include preserving identities, distribution, inverses, and the zero element. The map f:C->Z, where C is the set of complex numbers and Z is the set of integers, is an example of a homomorphism. Other examples include the identity map, zero map, and inclusion map, as well as maps such as f:Z->Z, where f(n) = 2n, and g:R[x]->R, where

Mar 30, 2012

Stephen88

I'm trying to see if the map f:C->Z,f(a+bi)=a is a homomorphism of rings.
Let x=a+bi and y=c+di...then f(x+y)=a+c=f(x)+f(y) but f(xy)=f((ac-db)+(ad+bc)i)=ac-db/= f(x)f(y)...so the map is not a homomorphism of rings.
Is this correct?

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Mar 30, 2012

Opalg

Gold Member

MHB

StefanM said:

Is this correct?

Yes. (Smile)

Related to Homomorphism of Rings & Is Map f:C->Z a Homo?

1. What is a homomorphism of rings?

A homomorphism of rings is a map between two rings that preserves the ring structure. This means that the map must preserve addition, multiplication, and the identity elements of both rings.

2. How is a homomorphism of rings different from an isomorphism?

While a homomorphism of rings only preserves the ring structure, an isomorphism also preserves the bijective property, meaning the map has both an inverse and a one-to-one correspondence between elements of the two rings.

3. What are the properties of a homomorphism of rings?

Some important properties of a homomorphism of rings include: it preserves the identity element, it distributes over addition and multiplication, it preserves inverses, and it preserves the zero element.

4. Is the map f:C->Z a homomorphism?

Yes, the map f:C->Z, where C is the set of complex numbers and Z is the set of integers, is a homomorphism. It preserves addition, multiplication, and the identity elements of both C and Z.

5. What are some examples of homomorphisms of rings?

Examples of homomorphisms of rings include the identity map, the zero map, and the inclusion map. Other examples include the map f:Z->Z, where f(n) = 2n, and the map g:R[x]->R, where g(p(x)) = p(0).