Need help about Quotient Rings (Factor Rings)

Thread starter AAQIB IQBAL
Start date Oct 9, 2012
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quotient Rings

In summary, a quotient ring is a smaller ring created by dividing a larger ring by an ideal. It differs from a regular ring in that its elements are equivalence classes of elements in the original ring and its operations may have different results due to the presence of the ideal. Ideals are subsets of a ring that are used to define the operations and determine which elements are considered zero in the quotient ring. Quotient rings have various applications in mathematics, such as in abstract algebra and coding theory. Some common examples of quotient rings include the integers mod n, polynomials with coefficients in a field, and the Gaussian integers.

Oct 9, 2012

AAQIB IQBAL

Suppose S is an Ideal of a Ring R
I want to verify the multiplication operation in the Factor Ring R/S which is
(S + a)(S + b) = (S + ab)
for this i Need to show that :
(S + ab) ≤ (S +a)(S + b)
please give me some idea about it
IT IS GIVEN AS DEFINITION IN THE BOOK I.N Herstein

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Oct 15, 2012

bpet

It might help to consider a concrete example such as S=5Z with a=2 and b=3.

1. What is a quotient ring?

A quotient ring, also known as a factor ring, is a mathematical structure that is created by taking a ring and dividing it by an ideal. The resulting quotient ring is a new ring that contains all the elements of the original ring that are not in the ideal.

2. How is a quotient ring different from a regular ring?

A quotient ring is different from a regular ring in that it is a smaller ring created by dividing a larger ring. The elements in a quotient ring are equivalence classes of elements in the original ring, while the elements in a regular ring are not necessarily related to each other. Additionally, the operations in a quotient ring are defined based on the operations in the original ring, but may have different results due to the presence of the ideal.

3. What is an ideal and how does it relate to quotient rings?

An ideal is a subset of a ring that satisfies certain properties. In the context of quotient rings, the ideal acts as a way to "identify" certain elements in the original ring that are considered equivalent in the quotient ring. The ideal is used to define the operations in the quotient ring and determines which elements are considered zero in the quotient ring.

4. How are quotient rings used in mathematics?

Quotient rings are used in many areas of mathematics, including abstract algebra, number theory, and algebraic geometry. They are useful for studying properties of rings and can help simplify complicated structures. Quotient rings also have applications in computer science and coding theory.

5. What are some common examples of quotient rings?

The integers mod n, denoted as Z/nZ, is a common example of a quotient ring. Other examples include polynomials with coefficients in a field, where the ideal is generated by a single polynomial, and the Gaussian integers, where the ideal is generated by the imaginary unit i.