# Algebra question - rings and ideals

• quasar987
In summary, the conversation discusses the definition of the ideal (I) in a commutative ring R and its relationship to the set A of polynomials with coefficients in I. The solution states that (I) is a subset of A and proves the other inclusion by showing that every element in A can be written as a sum of elements in (I).
quasar987
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[SOLVED] Algebra question - rings and ideals

## Homework Statement

Let R be a (nonzero) commutative ring with identity and I be an ideal of I. Denote (I) the ideal of R[x] generated by I. The book says that (I) is the set of polynomials with coefficients in I. Why is that?

## The Attempt at a Solution

Call A the set of polynomials with coefficients in I.

R is commutative and so is R[x], therefor (I) is simply given by

(I) = {a*p(x): a is in I and p(x) in R[x]}

So clearly (I) is a subset of A.

But for the other inclusion, given b_0+...+b_nx^n in A, we need to find an element a in I and a set {a_0,...,a_n} in R such that a*a_i = b_i for all i=1,...,n.

How is this achieved??

If b_0+...+b_nx^n is in A, then b_0, ..., b_n are all in I, and thus in (I). The latter is an ideal, so b_kx^k is certainly in (I), and so is the sum of such things.

Edit:
R is commutative and so is R[x], therefor (I) is simply given by

(I) = {a*p(x): a is in I and p(x) in R[x]}
I don't really see how this follows. I would instead prove that A is an ideal of R that contains I and hence contains (I) by definition.

Last edited:
Of course it does not follows! I was confused.

## 1. What is a ring in algebra?

A ring in algebra is a mathematical structure that consists of a set of elements and two binary operations, addition and multiplication. The operations follow certain rules, such as associativity and distributivity, and the set must contain a neutral element for each operation. Rings are used to study abstract algebraic structures and have many applications in mathematics and other fields.

## 2. What is the difference between a ring and a field?

A ring and a field are both algebraic structures, but they differ in the properties they possess. A field is a ring where every nonzero element has a multiplicative inverse, while a ring may not have this property. In other words, a field is a ring with additional properties that make it more similar to the real numbers.

## 3. What is an ideal in algebra?

An ideal in algebra is a subset of a ring that satisfies certain properties. It is similar to a subgroup in group theory, but with an additional requirement that the subset be closed under multiplication with any element from the ring. Ideals are used to study the structure of rings and can help determine the factors of polynomials.

## 4. How are rings and ideals related?

Ideals are closely related to rings, as they are subsets of rings with special properties. In fact, every ring has at least two ideals - the zero ideal, which contains only the additive identity, and the ring itself. Ideals can also be used to define important concepts in rings, such as prime and maximal ideals.

## 5. What are some applications of rings and ideals?

Rings and ideals have many applications in mathematics, physics, and computer science. They are used in abstract algebra to study the properties of algebraic structures, in number theory to factorize large numbers, and in cryptography to create secure communication systems. They also have applications in coding theory, algebraic geometry, and quantum mechanics.

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