Why is (2X,5) an ideal of Z[X]?

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Victor are discussing the properties of ideals in Z[X].In summary, DonAntonio and Victor are discussing how (2X,5) is an ideal of Z[X], even though it may not seem like it at first glance. This is because the element (5+2X)(7+X) is not in (2X,5) due to the "absorbance" property of ideals, contradicting the belief that all elements in (2X,5) must have an even linear coefficient. However, this is not always the case as shown by the example of 2x*1 + 5*x = 7x, which belongs to the ideal.
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Ad123q
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I have in my notes that (2X,5) is an ideal of Z[X], but I can't see why this can be so.

For example 5+2X is in (2X,5) and 7+X is in Z[X] but then

(5+2X)(7+X) =
= 35+5X+14X+2X^2
= 2X^2+19X+35.

19 is not divisible by 2 and so this element is not in (2X,5), contradicting the "absorbance" property of ideals.
 
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  • #2
Ad123q said:
I have in my notes that (2X,5) is an ideal of Z[X], but I can't see why this can be so.

For example 5+2X is in (2X,5) and 7+X is in Z[X] but then

(5+2X)(7+X) =
= 35+5X+14X+2X^2
= 2X^2+19X+35.

19 is not divisible by 2 and so this element is not in (2X,5), contradicting the "absorbance" property of ideals.



You seem to believe that any element in [itex](2x,5)[/itex] must have an even lineal coefficient, but this is wrong: the 5 there can multiply some x-coeff. of some

pol. and added to the even coefficient in the other factor we get an odd coef.

For example, the element [itex]2x\cdot 1 + 5\cdot x = 7x[/itex] belongs to the ideal...

DonAntonio
 

1. What is a simple ring?

A simple ring is a ring that has no non-trivial ideals, meaning that it has no proper subsets that are closed under multiplication and addition. In other words, every element in a simple ring has a multiplicative inverse and the only ideals are the trivial ones: the entire ring and the zero ideal.

2. What is an ideal in a ring?

An ideal in a ring is a subset of the ring that is closed under addition and multiplication by any element of the ring. In other words, if you take any element from the ring and multiply it by an element from the ideal, the result will still be in the ideal. Ideals are similar to normal subgroups in group theory.

3. How is an ideal different from a subring?

An ideal is a special type of subring that is closed under multiplication by any element of the original ring, while a subring is simply a subset of the original ring that is itself a ring under the same operations. In other words, all ideals are subrings, but not all subrings are ideals.

4. What is an ideal quotient?

An ideal quotient is a quotient ring formed by dividing a ring by an ideal. This means that all elements in the ideal are considered to be equivalent to the zero element in the quotient ring. Ideal quotients are used to study the structure and properties of a ring by breaking it down into smaller, more manageable parts.

5. How are ideals used in mathematics?

Ideals are used in many areas of mathematics, including algebra, number theory, and algebraic geometry. They are particularly useful in understanding the structure and properties of rings and can help simplify complicated problems. Additionally, ideals have applications in areas such as coding theory, cryptography, and the study of symmetry in mathematics.

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