# Zero divisors of an endomorphism ring

• I
• Danijel
In summary, the conversation discussed the Abelian group (A,+) and the ring E=End(A,A) of endomorphisms on the set A. The binary operations + and * were defined as (f+g)(x)=f(x) + g(x) and (f*g)=f∘g. The speaker mentioned trying to find zero divisors in this ring, but was unable to come up with an example. They then asked about nilpotent endomorphisms/matrices as a potential solution.
Danijel
Let (A,+) be an Abelian group. Consider the ring E=End(A,A) of endomorphisms on the set A, with binary operations +, and *, where (f+g)(x)=f(x) + g(x), and (f*g)=f∘g.
I have tried to find zero divisors in this ring, but I just couldn't come up with an example.

Danijel said:
Let (A,+) be an Abelian group. Consider the ring E=End(A,A) of endomorphisms on the set A, with binary operations +, and *, where (f+g)(x)=f(x) + g(x), and (f*g)=f∘g.
I have tried to find zero divisors in this ring, but I just couldn't come up with an example.
Have you heard about nilpotent endomorphisms / matrices?

## 1. What are zero divisors of an endomorphism ring?

Zero divisors of an endomorphism ring are elements that, when multiplied by any other element in the ring, result in the product being equal to zero. In other words, they are elements that have no multiplicative inverse in the ring.

## 2. What is the significance of zero divisors in an endomorphism ring?

Zero divisors play a crucial role in determining the structure and properties of an endomorphism ring. For example, if a ring has zero divisors, it cannot be a field. Additionally, the presence of zero divisors can affect the commutativity and invertibility of elements in the ring.

## 3. How are zero divisors identified in an endomorphism ring?

In an endomorphism ring, zero divisors can be identified by multiplying each element in the ring by another element and checking if the resulting product is equal to zero. If there exists at least one such product, then the ring has zero divisors.

## 4. Can an endomorphism ring have only one zero divisor?

Yes, it is possible for an endomorphism ring to have only one zero divisor. In fact, this is the case for the ring of integers modulo n, where n is not a prime number. In this case, the only zero divisor is the element n itself.

## 5. What are the implications of a ring having no zero divisors?

If a ring has no zero divisors, it is called an integral domain. This has several implications, such as all non-zero elements having multiplicative inverses, making the ring a field. Additionally, integral domains have unique factorization, making them useful in algebraic number theory and cryptography.

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