Is Every Prime Ideal Maximal in a Commutative Ring?

Understanding the relationship between prime ideals and maximal ideals in a commutative ring is a fundamental concept in algebra, particularly in ring theory. The answer to the question of whether every prime ideal is also a maximal ideal often depends on the specific structure and properties of the ring in question.

Definitions

Before we delve into the relationship between prime and maximal ideals, it's important to establish the definitions of these concepts:

Prime Ideal

An ideal (P) in a ring (R) is called a prime ideal if whenever the product of two elements (a cdot b) is in (P), then at least one of (a) or (b) must be in (P). Formally, (P) is prime if (a cdot b in P) implies (a in P) or (b in P).

Maximal Ideal

An ideal (M) in a ring (R) is called a maximal ideal if it is proper (i.e., not equal to the whole ring (R)) and if there are no other ideals strictly between (M) and (R). Formally, (M) is maximal if whenever (M subseteq I subseteq R), then (I M) or (I R).

Relationship Between Prime and Maximal Ideals

Maximal (Rightarrow) Prime: In any commutative ring with unity, every maximal ideal is prime. This is because if an ideal (M) is maximal and (a cdot b in M) but neither (a) nor (b) is in (M), then we could form a larger ideal that contradicts the maximality of (M).

Prime ( Rightarrow) Maximal: The converse is not true in general; not all prime ideals are maximal. A prime ideal might not be maximal in some rings, particularly in non-field rings.

Examples

Example in the Ring of Integers (mathbb{Z})

In the ring of integers (mathbb{Z}):

The ideal (5) is maximal because (mathbb{Z}/5 cong mathbb{Z}_5) is a field, and maximal ideals correspond to quotient rings that are fields. The ideal (0) is prime because (mathbb{Z}) is a domain; if (a cdot b 0) in (mathbb{Z}), then either (a 0) or (b 0). However, (0) is not maximal because there are many ideals between (0) and (mathbb{Z}), such as (2, 3, 4,) and so on.

Example in Polynomial Rings

Consider the ring (R mathbb{Z}[x]):

The ideal (2x) generated by (2) and (x) is a prime ideal because (mathbb{Z}[x]/2x cong mathbb{Z}_2[x]), which is an integral domain. However, (2x) is not maximal because the quotient ring is not a field.

When Are Prime Ideals Maximal?

Fields

In a field, the only prime ideal is (0) and it is also maximal because a field has no proper non-zero ideals.

Principal Ideal Domains (PIDs)

In a PID every non-zero prime ideal is maximal. However, the zero ideal (0) in a PID like (mathbb{Z}) is prime but not maximal.

Conclusion

While every maximal ideal is prime, not every prime ideal is maximal. The relationship between prime and maximal ideals depends on the specific structure of the ring being considered. Understanding these relationships is crucial for delving deeper into ring theory and abstract algebra.