What are the ideals of the ring Z[x]/<2, x^3 + 1>?

In summary, ideals of polynomial rings are subsets of the ring that satisfy certain properties and are used to generalize the concept of divisibility. They are closely related to factorization and can have more than one generator. Ideals can contain zero divisors and are used in algebraic geometry to study solutions to polynomial equations.
  • #1
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I would like help to get started on the following problem:

Determine all the ideals of the ring \(\displaystyle \mathbb{Z}[x]/<2, x^3 + 1> \)

Appreciate some guidance.

Peter
 
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  • #2
Have a look at problem 6:

http://livetoad.org/Courses/Documents/84dd/Exams/answers_exam_1.pdf
 
  • #3
Fernando Revilla said:
Have a look at problem 6:

http://livetoad.org/Courses/Documents/84dd/Exams/answers_exam_1.pdf

Thanks Fernando, most helpful

Peter
 

FAQ: What are the ideals of the ring Z[x]/<2, x^3 + 1>?

What are ideals of polynomial rings?

Ideals of polynomial rings are subsets of the ring that satisfy certain properties. In particular, they must be closed under addition and multiplication by any element in the ring. They are used to generalize the concept of divisibility in polynomial rings.

How are ideals of polynomial rings related to factorization?

Ideals of polynomial rings are closely related to factorization because they represent the set of all polynomials that are divisible by a given polynomial. This means that the elements of an ideal can be thought of as the factors of a polynomial. In fact, the ideal generated by a polynomial is the set of all multiples of that polynomial.

Can ideals of polynomial rings have more than one generator?

Yes, ideals of polynomial rings can have more than one generator. This means that the ideal is generated by a set of polynomials, rather than just a single polynomial. In general, an ideal can have infinitely many generators.

What is the relationship between ideals of polynomial rings and zero divisors?

Ideals of polynomial rings can contain zero divisors, but not all zero divisors are contained in ideals. A zero divisor is an element that, when multiplied by another element, results in 0. In polynomial rings, an ideal that contains a zero divisor will also contain all of its multiples.

How are ideals of polynomial rings used in algebraic geometry?

Ideals of polynomial rings are used in algebraic geometry to study the solutions to polynomial equations. In particular, the set of common zeros of a set of polynomials in a polynomial ring can be thought of as the set of solutions to a system of polynomial equations. Ideals are used to represent these sets, allowing for the use of algebraic techniques to study geometric objects.

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