Determine all the ideals of the ring Z

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In summary, the ideals of the ring Z[x]/(2,x^3+1) are all multiples of x^3+1 with their free term being an even number. They can also be found by taking the kernel of a homomorphism or by lifting an ideal from Z[x] to Z[x]/(2,x^3+1). Another approach is to directly write out all the elements of the ring and work out its ideals from there.
  • #1
b0mb0nika
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determine all the ideals of the ring Z[x]/(2,x^3+1)

i'm a bit confused b/c this is a quotient ring.
would the ideals be all the polynomials which are multiples of x^3+1, with their free term an even number ?
 
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  • #2
I'm not sure what you're saying, but it sounds wrong.

Here are a couple of ideas that might help:

(1) Recall that any ideal is the kernel of some homomorphism...

(2) You can lift any ideal of Z[x]/(2,x^3+1) to an ideal of Z[x]...

(3) Z[x]/(2,x^3+1) is a rather small ring...
 
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  • #3
the ideals of a quotient ring are those ideals of the original ring that contain the ideal in the bottom of the quotient, so you are looking for all ideals of Z[X] that contain both X^3 +1 and 2.
 
  • #4
i was thinking kind of the same thing as mathwonk.. but I'm not sure how to write those ideals..
if they contain X^3+1 and 2.. wouldn't they have to be multiples of them ?
 
  • #5
Does (4) contain 2?

Whatever method you try, I recommend also trying my third suggestion -- the ring is small, so you can explicitly write all of the elements of the ring, and directly work out all of its ideals.
 
  • #6
Hurkyl's suggestions are always valuable.

Also, remember the elements of the smallest ideal containing u and v, consists of all linear combinations of form au+bv, with a,b, in the ring.
 

FAQ: Determine all the ideals of the ring Z

What is a ring?

A ring is a mathematical structure consisting of a set of elements along with two binary operations, addition and multiplication, that satisfy certain axioms. The most common example of a ring is the set of integers (Z) with the operations of addition and multiplication.

What are ideals in a ring?

Ideals in a ring are subsets of the ring that satisfy specific properties. In particular, an ideal is a subset that is closed under addition and multiplication by elements of the ring. They are important in abstract algebra and have applications in various areas of mathematics.

How do you determine all the ideals of the ring Z?

To determine all the ideals of the ring Z, we can use the fact that every ideal of Z is generated by a single integer. Therefore, to find all the ideals, we need to consider all possible integers as generators and check which ones satisfy the properties of an ideal.

What are some examples of ideals in the ring Z?

Some examples of ideals in the ring Z are the zero ideal (consisting only of the element 0), the principal ideals generated by individual integers (such as the ideal (2) consisting of all multiples of 2), and the entire ring Z itself.

What is the significance of determining all the ideals of the ring Z?

Determining all the ideals of the ring Z allows us to better understand the structure of this important mathematical object. It also has applications in fields such as number theory and algebraic geometry, where the properties of ideals are used to prove theorems and solve problems.

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