Exploring Prime and Maximal Ideals in Z[x]: The Case of x^2 - 1

In the realm of abstract algebra, the behavior of ideals in polynomial rings and their properties like primality and maximality are fundamental concepts. This article delves into the specifics of the ideal generated by the polynomial x2 - 1 in the polynomial ring Z[x]. We will prove two key properties of this ideal: that it is a prime ideal but not a maximal ideal.

Step 1: Proving the Ideal Is a Prime Ideal in Z[x]

Let's start by considering the ideal Langle x2 - 1 Rangle in the polynomial ring Z[x]. An ideal I in a ring R is called a prime ideal if for any elements a, b in R, whenever ab is in I, then at least one of a or b must be in I.

Show that x2 - 1 Factors Over a Different Field

Consider the polynomial x2 - 1. This polynomial factors as (x - 1)(x 1) over the field of real numbers. However, over the complex field C, it can be properly factored as (x - i)(x i).

Quotient Ring is isomorphic to the Ring of Gaussian Integers

Consider the quotient ring Z[x]/Langle x2 - 1 Rangle. By the First Isomorphism Theorem, this quotient ring is isomorphic to the ring of Gaussian integers Z[i]. The Gaussian integers are the set of complex numbers with integer coefficients: {a bi | a, b ∈ Z}.

The ring Z[i] is an integral domain, meaning it has no zero divisors. Therefore, if fx and gx in Z[x] are polynomials such that their product fxgx is in the ideal Langle x2 - 1 Rangle, then we must have either fx or gx in the ideal Langle x2 - 1 Rangle.

Step 2: Proving the Ideal Is Not a Maximal Ideal in Z[x]

An ideal I in a ring R is called maximal if the only ideals containing I are I itself and R. To show that Langle x2 - 1 Rangle is not a maximal ideal in Z[x], we need to find a proper ideal in Z[x] that contains Langle x2 - 1 Rangle.

Quotient Ring is equivalent to Z[i]

Recall that Z[x]/Langle x2 - 1 Rangle ? Z[i]. The ring of Gaussian integers Z[i] is itself a ring with no zero divisors and contains proper ideals, which are not the whole ring itself. For example, the principal ideal ?2? in Z[i] is a proper ideal that contains Langle x2 - 1 Rangle. This is because 2 can be seen as a constant polynomial in Z[x], which generates a proper ideal in Z[x].

Conclusion

In conclusion, the ideal Langle x2 - 1 Rangle in the polynomial ring Z[x] is a prime ideal because it corresponds to an integral domain in Z[i]. It is not a maximal ideal because it can be contained in a proper ideal in Z[i] such as ?2?.

This proof exemplifies the rich interplay between polynomial rings and their associated quotient rings, highlighting the prime and maximal properties of ideals. Such knowledge is foundational in understanding algebraic structures and their applications in various fields of mathematics.