# A question about kernels and commutative rings

• Artusartos
In summary, a kernel in mathematics is a set of elements that map to the identity element under a given map or operation. Kernels are significant in commutative rings because they help determine the structure of the ring and can be used to define important concepts like ideals and quotient rings. To calculate the kernel of a homomorphism, one must determine the operation or map and then find the set of elements that map to zero. Kernels are closely related to quotient rings, as the quotient ring is defined by factoring out the elements in the kernel. Multiple kernels can exist for a commutative ring, but there may also be cases where a unique kernel exists for a specific operation or map.
Artusartos

## Homework Statement

http://www.math.wvu.edu/~hjlai/Teaching/Math541-641/Math_541_HW_4_2004.pdf

The part that I don't understand is...Describe ker ϵ in terms of roots of
polynomials. Does this just mean "What is the kernel of ϵ?"

## The Attempt at a Solution

Is my answer correct (I think it's a bit different from the answer in the doc.)...if not, can you tell me why it is wrong?

Ker ϵ = {f(x) in R[x] : ϵ(f(x)) = 0} = $$\{f(x) \in R[x]: f(x) = 0 +a_1x + a_2x^2+ ... + a_nx^n\}$$

## What is a kernel in mathematics?

A kernel in mathematics refers to the set of elements that map to the identity element under a given map or operation. In other words, it is the set of elements that are mapped to zero under a homomorphism.

## What is the significance of kernels in commutative rings?

In commutative rings, kernels play an important role in understanding the structure of the ring. They can help determine whether a map or operation is injective or surjective, and can also be used to define important concepts like ideals and quotient rings.

## How do you calculate the kernel of a homomorphism?

To calculate the kernel of a homomorphism, you must first determine the operation or map that the homomorphism is defined on. Then, you can use the definition of a kernel to find the set of elements that map to zero under the given operation or map.

## What is the relationship between kernels and quotient rings?

The kernel of a homomorphism is closely related to the concept of quotient rings. In fact, the kernel of a homomorphism is always an ideal of the ring, and the quotient ring is defined as the ring obtained by factoring out the elements in the kernel. This relationship is often used in algebraic structures to simplify calculations and proofs.

## Can a commutative ring have multiple kernels?

Yes, a commutative ring can have multiple kernels. In fact, any map or operation can have multiple kernels, as the definition of a kernel only requires that the elements in the set map to zero under the given operation or map. However, in some cases, there may be a unique kernel for a particular operation or map.

• Calculus and Beyond Homework Help
Replies
14
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Linear and Abstract Algebra
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
24
Views
2K