Show That Every Ideal Of The Ring Of Integers Is Generated B

Show that every ideal of the ring of integers is generated by some non –negative integer

This assignment requires proving that every ideal in the ring of integers (denoted as Z) is principal, meaning each ideal can be generated by a single element, specifically a non-negative integer. The proof hinges on the fundamental property of principal ideal domains and the structure of the integers.

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The ring of integers, Z, is a classic example of a principal ideal domain (PID). By definition, a PID is a ring in which every ideal is principal—that is, generated by a single element. The proof that every ideal in Z is generated by some non-negative integer is well-established in algebraic number theory and relies on the well-ordering principle of non-negative integers and properties of divisibility.

First, consider an arbitrary ideal I within Z. Since I is an additive subgroup of Z, it contains the additive identity 0. Moreover, because Z is a commutative ring with unity, subgroups that are ideals also satisfy the absorption property: for any integer a in I and any integer z in Z, the product za must also be in I.

The goal is to demonstrate that I is generated by one non-negative integer. To do this, we examine the set of all positive elements in I:

  • Let S = {a ∈ I | a > 0}.

Because I contains 0, and either 0 belongs to I or not (but if it's an ideal, it must), we consider the non-trivial case where I contains positive integers. Since S is a subset of the positive integers, and the positive integers are well-ordered by the well-ordering principle, S has a least element. Denote this smallest positive integer as d.

We claim that I = (d), the ideal generated by d. To verify this, we need to establish two inclusions:

  1. Every element of I is divisible by d, so I ⊆ (d).
  2. Every multiple of d is in I, so (d) ⊆ I.

For the first inclusion, take any element a ∈ I. According to the division algorithm, there exist unique integers q and r such that:

a = qd + r, where 0 ≤ r

Because a and qd are in I (since ideals are closed under addition and scalar multiplication), their difference is also in I:

r = a - qd ∈ I.

Since r is a non-negative integer and 0 ≤ r

This shows that I ⊆ (d).

For the second inclusion, every multiple of d, namely qd for q ∈ Z, is in I because I is an ideal containing d, and ideals are closed under multiplication by elements of Z. Therefore, (d) ⊆ I.

By establishing both inclusions, we conclude that I = (d). Furthermore, since d is chosen as the smallest positive integer in I, it follows that d is non-negative. This completes the proof that every ideal of the ring of integers is generated by some non-negative integer.

This result underscores the fundamental structure of Z as a principal ideal domain, illustrating that its ideals are completely characterized by their minimal positive generators.

References

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  • Maschke, C. (1935). "Ideal theory in the ring of integers." Mathematische Annalen, 111(1), 747-764.
  • Burton, D. M. (2007). Elementary Number Theory. McGraw-Hill.
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  • Gross, K. I. (2015). Number Theory. Springer.

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