Rings and Homomorphism example

In summary, the function f cannot be a homomorphism because the equation f(a+b)=f(a)f(b) does not always hold.
  • #1
dancergirlie
200
0

Homework Statement


Give an example of a ring R and a function f: R---->R such that f(a+b)=f(a)f(b) for all a,b in R. and f(a) is not the zero element for all a in R. Is your function a homomorphism?


Homework Equations


Let R and S be rings. A function f:R----->S is said to be a homomorphism if
f(a+b)=f(a) + f(b) and f(ab)=f(a)f(b) for all a,b in R


The Attempt at a Solution



Not really sure where to start here,
I was thinking about using Zn as my ring, perhaps with n as a prime number, so that way f(a) wouldn't be zero for any a. but i don't know what my function would be to satisfy that. Any help would be greatly appreciated =)
 
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  • #2
can the ring be a division ring?
 
  • #3
I don't think so, because I don't think we've learned about division rings yet lol
 
  • #4
dancergirlie said:
but i don't know what my function would be to satisfy that.
By "that", do you mean f(a+b)=f(a)f(b)?

That's an easy problem -- if you want f to satisfy that, and if you don't know a particular value of f, then the equation tells you how to compute it!
 
  • #5
I don't think that is what I need to do. It asks to give a specific ring and function that fits those specifications. Like for ring Z,
f(x)=x^2, but that doesn't work.
 
  • #6
If you're going to look for a ring and a function for which f(a+b)=f(a)f(b) is satisfied, you might as well use that equation to help you figure out what f should be. *shrug*
 
  • #7
I'm not sure if I follow you here, should I just plug in values from the ring and see if I can notice a pattern? I am just completely lost with this problem, I don't even know which ring I should use.
 
  • #8
Even if you don't notice a pattern, it will help you get started narrowing down a few specific values of f.

What ring to use? You had a few ideas you wanted to try, right? Do those! Or... you could start with values that every ring has. (e.g. 0, 1, 2...)
 
  • #9
I seriously have been looking at this for at least an hour now and have made no progress... I have tried the ring of even numbers, just Z, the ring Z5. I just don't know what to do, I'm not exactly an expert at the whole rings thing yet.
 
  • #10
What kind of functions do you know that turn addition into multiplication (or vice versa)? That's what the question is asking you.
 
  • #11
Thank you so much! I didn't even think of the exponential functions. I got that my function was obviously not homomorphic since f(a+b) did not equal f(a)+f(b) and f(ab) was not equal to f(a)f(b). I think I was just too tired at the moment to think of that function. Thanks for the help everyone!
 
  • #12
That isn't quite the right reasoning for why such a map cannot be an homomorphism. Why can't f(a)f(b) equal f(a)+f(b)? Just because the expressions look different doesn't mean that they are, really. Just take the possibly illegal case of the ring with one element :0.

But the question is trivial since it provides a reason why f can't be a homomorphism in its own statement: you are told that f(a) is never 0, and homomorphisms send 0 to 0. Note that the question precludes the example of the ring with one element from being considered.
 

What is a ring?

A ring is a mathematical structure consisting of a set of elements, along with two binary operations (usually addition and multiplication) that satisfy certain properties. Examples of rings include the set of integers, real numbers, and matrices.

What is a homomorphism?

A homomorphism is a function between two algebraic structures that preserves their operations. In other words, if we apply the operation on two elements in the first structure, the result will be the same as applying the operation on the corresponding elements in the second structure.

What is an example of a ring?

An example of a ring is the set of integers, denoted as ℤ. The set of integers is closed under addition and multiplication, and satisfies the properties of associativity, commutativity, and distributivity. It also contains an identity element (0) and every element has an additive inverse.

What is an example of a homomorphism?

An example of a homomorphism is the function f: ℤ → ℤ where f(x) = 2x. This function preserves the operations of addition and multiplication, as f(x + y) = f(x) + f(y) and f(xy) = f(x)f(y). Therefore, f is a homomorphism from the ring of integers to itself.

How are rings and homomorphisms used in science?

Rings and homomorphisms have many applications in science, particularly in areas such as abstract algebra, number theory, and cryptography. They can be used to study the properties of different mathematical structures and to solve complex problems in various fields of science and engineering.

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