Finding Elements of a Quotient Ring

In summary, the conversation is about finding elements of a quotient ring. The person is seeking advice on how to approach this task and suggests taking the representative with the smallest degree. They also mention the importance of stating the question clearly and asking for help in the appropriate forums.
  • #1
Soccer4822
1
0
Hello all, first time to the site and its very helpful! I wish I would have found it sooner.
I am stuck on quotient rings. Here is my question..

How do I find elements of a quotient ring?

It asks me to list all elements of a quotient ring.

Anybody have any ideas how i can find them? :confused:
 
Last edited:
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  • #2
You'll have to state the question.
And by the way, I think these questions (that is, HW questions) should be asked several forums above.
 
  • #3
In your quotient ring, all the elements look like p(x)+I. Now you know for a given element in the quotient ring, you can take many different p(x)'s as it's representative. Try to take the one with the smallest degree.

-I see you've edited while I was replying, so the above may look strange to others.
 

Related to Finding Elements of a Quotient Ring

1. What is a quotient ring?

A quotient ring is a mathematical structure that is formed by taking a ring and "quotienting out" a subset of elements. This means that certain elements are considered equivalent and are thus combined into a single element in the quotient ring.

2. How do you find elements of a quotient ring?

To find elements of a quotient ring, you first need to define the subset of elements that will be considered equivalent. Then, for each element in the original ring, you divide it by the subset and take the remainder. The resulting remainders will be the elements of the quotient ring.

3. What is a coset in a quotient ring?

A coset in a quotient ring is a set of elements that are equivalent to each other and are obtained by adding a fixed element from the subset being "quotiented out." It is essentially a group of elements that are related by the subset.

4. How is the quotient ring related to the original ring?

The quotient ring is a smaller ring that is formed from the original ring by "quotienting out" a subset of elements. This means that the elements of the quotient ring are related to the elements of the original ring, but they may be combined or represented differently.

5. What are some applications of quotient rings?

Quotient rings have various applications in mathematics, particularly in abstract algebra. They can be used to study the structure of rings, to solve polynomial equations, and to understand the properties of groups and fields. They also have practical applications in areas such as coding theory and cryptography.

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