Problem 17 - Algebra: Pure and Applied

Question

Show that in an integral domain \(D\), a nonzero element \(p \in D\) is prime if and only if \(\langle p\rangle\) is a prime ideal in \(D\).

Step-by-Step Solution

Answer

In an integral domain, an element is prime if its principal ideal is prime.

Step 1: Understanding the Definitions

An integral domain is a commutative ring with no zero divisors. An element \( p \in D \) is prime if whenever \( p \) divides a product \( ab \), it divides at least one of \( a \) or \( b \). An ideal \( \langle p \rangle \) is a prime ideal if whenever a product \( ab \in \langle p \rangle \), then either \( a \in \langle p \rangle \) or \( b \in \langle p \rangle \).

Step 2: Showing Prime Element Implies Prime Ideal

Assume \( p \) is a prime element. If a product \( ab \in \langle p \rangle \), then there exists some \( d \in D \) such that \( ab = pd \). Since \( p \) is prime and divides \( ab \), \( p \) must divide either \( a \) or \( b \). Therefore, either \( a \in \langle p \rangle \) or \( b \in \langle p \rangle \), which means \( \langle p \rangle \) is a prime ideal.

Step 3: Showing Prime Ideal Implies Prime Element

Assume \( \langle p \rangle \) is a prime ideal. If \( p \mid ab \), then \( ab = pd \) for some \( d \). Since \( ab \in \langle p \rangle \), and \( \langle p \rangle \) is a prime ideal, \( a \in \langle p \rangle \) or \( b \in \langle p \rangle \). Thus, \( p \mid a \) or \( p \mid b \), proving \( p \) is prime.

Step 4: Conclusion

Since both directions of implication have been established, a nonzero element \( p \in D \) is prime if and only if \( \langle p \rangle \) is a prime ideal.

Key Concepts

Integral Domain

An integral domain is a mathematical structure that combines the features of a commutative ring with the absence of zero divisors, making it a highly consistent environment for solving problems involving division. To understand it, first know that:

• A **commutative ring** is a set equipped with two operations: addition and multiplication, where the multiplication operation is commutative, meaning for any two elements \( a \) and \( b \), \( ab = ba \).
• An integral domain takes it a step further by ensuring there are no zero divisors. This means for any non-zero elements \( a \) and \( b \) in the domain, the product \( ab e 0 \).

This property is crucial because it allows division, or cancellation, in the algebraic sense without issue, making arithmetic within integral domains more predictable and similar to the familiar properties of the integers.

Prime Ideal

Prime ideals are a fascinating topic in ring theory, serving as a cornerstone for understanding various algebraic structures. To digest this, let's break down some key points:

• In a commutative ring, an **ideal** is a subset that absorbs multiplication by any element of the ring but doesn't need to include all ring elements.
• A prime ideal \( \langle p \rangle \) has an additional intriguing property: if a product of two elements \( ab \) is in the ideal, then at least one of the elements \( a \) or \( b \) must itself be in the ideal.

This characteristic of prime ideals makes them analogous to prime numbers in integers, which similarly divide products in a specific way. Prime ideals generalize this property to more abstract rings, forming a conceptual bridge in algebra.

Commutative Ring

Commutative rings serve as a foundational structure in algebra, underpinning more complex concepts like fields and integral domains. They exhibit several critical traits:

• A commutative ring is equipped with two binary operations—addition and multiplication—that satisfy common arithmetic rules. Addition forms an abelian group and multiplication is associative and commutative.
• An important aspect is the presence of a multiplicative identity (often denoted by 1), although not all definitions require this as a component.

The significance of commutative rings lies in their ability to model various algebraic systems, including polynomial rings and function rings, where understanding the behaviour of operations is crucial. They provide a platform for exploring deeper mathematical relationships and properties.

Zero Divisors

Zero divisors present an important concept in understanding the structure of rings and integral domains in particular. Here's what you need to know:

• A zero divisor is an element of a ring that provides a nonzero product when multiplied with another element, yet the result is zero.
• In an equation, if \( a \cdot b = 0 \) and both \( a \) and \( b \) are non-zero, then \( a \) and \( b \) are zero divisors.

Zero divisors complicate arithmetic because they break the typical rule that only zero can result from a multiplication if at least one operand is zero. Understanding zero divisors is crucial in differentiating between normal rings and integral domains, the latter being free from zero divisors, ensuring reliability in mathematical operations and enabling concepts like cancellative division.