# Find all ring homomorphisms from 3Z to Z?

• mathgirl313
In summary, there are two ring homomorphisms from 3Z to Z, the zero homomorphism and the identity homomorphism.
mathgirl313

## Homework Statement

Find all ring homomorphisms from 3Z to Z, where 3Z are the integers that are of multiple 3.

## The Attempt at a Solution

So 3Z is cyclic so σ(3) is sufficient to look. Now all of the other examples have finite groups, so |σ(a)| divides the |a| in the domain. Or, the homomorphisms is define with Z as the domain, and then a=a^2. I understand these examples, but then don't know how to start this one...

<3> generates the group, so I know I want to consider it. But if σ(3)=a, but then what? Other examples are a=σ(1)=σ(1*1)=a^2, but that property doesn't hold here, and 1 is not in3Z. I know I'm missing something simple, just can't quite figure out what property I need to put my finger on to make that first step..

Thanks for any help!

mathgirl313 said:

## Homework Statement

Find all ring homomorphisms from 3Z to Z, where 3Z are the integers that are of multiple 3.

## The Attempt at a Solution

So 3Z is cyclic so σ(3) is sufficient to look. Now all of the other examples have finite groups, so |σ(a)| divides the |a| in the domain. Or, the homomorphisms is define with Z as the domain, and then a=a^2. I understand these examples, but then don't know how to start this one...

<3> generates the group, so I know I want to consider it. But if σ(3)=a, but then what? Other examples are a=σ(1)=σ(1*1)=a^2, but that property doesn't hold here, and 1 is not in3Z. I know I'm missing something simple, just can't quite figure out what property I need to put my finger on to make that first step..

Thanks for any help!

You've got the right starting point. So σ(3)+σ(3)+σ(3)=σ(3+3+3)=σ(9)=σ(3*3)=σ(3)*σ(3). So?

Dick said:
You've got the right starting point. So σ(3)+σ(3)+σ(3)=σ(3+3+3)=σ(9)=σ(3*3)=σ(3)*σ(3). So?

So 3a=a^2 if a=σ(3). So the possible homomorphisms of a will have that property. And it should them just fail out like the other examples.

mathgirl313 said:
So 3a=a^2 if a=σ(3). So the possible homomorphisms of a will have that property. And it should them just fail out like the other examples.

If a solves 3a=a^2 then it's a homomorphism. You still need to describe what the homomorphisms are.

Last edited:
Dick said:
If a solves 3a=a^2 then it's a homomorphism. You still need to describe what the homomorphisms are.

Then 3a - a^2 = 0, and 3Z has no zero divisors, so factoring the equation gives a=0 or a=3.
Then, if σ(3)=a=0, then σ(x)=0 for all x in 3Z, and σ is the zero homomorphism.
If σ(3)=a=3, then σ(x)=x for all x in 3Z, and σ is the identity homomorphism.

Hence there there two ring homomorphism for 3Z to Z.

mathgirl313 said:
Then 3a - a^2 = 0, and 3Z has no zero divisors, so factoring the equation gives a=0 or a=3.
Then, if σ(3)=a=0, then σ(x)=0 for all x in 3Z, and σ is the zero homomorphism.
If σ(3)=a=3, then σ(x)=x for all x in 3Z, and σ is the identity homomorphism.

Hence there there two ring homomorphism for 3Z to Z.

Seems fine to me.

## 1. What is a ring homomorphism?

A ring homomorphism is a function between two rings that preserves the structure and operations of the rings. This means that if we have two elements a and b in one ring, their sum and product will be mapped to the same sum and product in the other ring.

## 2. What is 3Z and Z?

3Z is the set of all integers that are divisible by 3, and Z is the set of all integers. In other words, 3Z is a subset of Z.

## 3. How many ring homomorphisms are there from 3Z to Z?

There are infinitely many ring homomorphisms from 3Z to Z. This is because for any integer n, we can define a ring homomorphism f: 3Z -> Z such that f(x) = nx for all x in 3Z.

## 4. How do you find all ring homomorphisms from 3Z to Z?

To find all ring homomorphisms from 3Z to Z, we can follow these steps:

1. Start by defining a ring homomorphism f: 3Z -> Z.
2. Since we know that f(x) = nx for all x in 3Z, we can choose any integer n as the value of f(3).
3. Next, we need to check if this choice of n satisfies the ring homomorphism property for all elements in 3Z. If it does, then f is a valid ring homomorphism from 3Z to Z.
4. Repeat this process for different values of n to find all possible ring homomorphisms.

## 5. What are some applications of ring homomorphisms?

Ring homomorphisms have many applications in different areas of mathematics, such as abstract algebra, number theory, and algebraic geometry. They are also used in cryptography, coding theory, and computer science. Specifically, in the context of 3Z and Z, ring homomorphisms can be used to study divisibility and remainders of integers.

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