Find the zero divisors and the units of ##\mathbb Z[X]/<X^3>##

In summary, we are asked to find the zero divisors and units of the quotient ring ##\mathbb Z[X]/<X^3>##. After some attempts, the elements ##a=X+<X^3>## and ##b=X^2+<X^3>## are identified as zero divisors. It is suspected that these are the only zero divisors, but further confirmation is needed. As for the units, there is currently no clear approach. Additional information or suggestions would be appreciated.
  • #1
mahler1
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Homework Statement

Find the zero divisors and the units of the quotient ring ##\mathbb Z[X]/<X^3>##


The attempt at a solution

If ##a \in \mathbb Z[X]/<X^3>## is a zero divisor, then there is ##b \neq 0_I## such that ##ab=0_I##. I think that the elements ##a=X+<X^3>## and ##b=X^2+<X^3>## are zero divisors because we have

##ab=XX^2+<X^3>=X^3+<X^3>=<X^3>##.

I couldn't think of any other divisors so I suspect these two are the only ones. Am I correct? If that is the case, how could I show these are the only zero divisors?

As for the units I don't know what to do. Any suggestions would be appreciated.
 
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Related to Find the zero divisors and the units of ##\mathbb Z[X]/<X^3>##

1. What is a zero divisor in the context of ##\mathbb Z[X]/##?

A zero divisor in the context of ##\mathbb Z[X]/## is an element that, when multiplied by another element in the quotient ring, results in the zero element. In other words, it is an element that does not have a multiplicative inverse.

2. How do I find the zero divisors of ##\mathbb Z[X]/##?

To find the zero divisors of ##\mathbb Z[X]/##, you can first find all elements in the quotient ring and then multiply each one by every other element. Any resulting product that is equal to zero is a zero divisor.

3. What are the units of ##\mathbb Z[X]/##?

The units of ##\mathbb Z[X]/## are the elements that have a multiplicative inverse. In this quotient ring, the only units are the elements that are relatively prime to ##X^3##, or in other words, the constant terms.

4. Can a zero divisor be a unit in ##\mathbb Z[X]/##?

No, a zero divisor cannot be a unit in ##\mathbb Z[X]/##. This is because a unit must have a multiplicative inverse, but a zero divisor does not have one.

5. Are there any other properties of the units and zero divisors in ##\mathbb Z[X]/##?

Yes, in fact, the units and zero divisors in ##\mathbb Z[X]/## have some interesting properties. For example, all zero divisors are nilpotent (meaning they become zero after a certain number of multiplications) and the units form a group under multiplication.

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