Proving a Commutative Ring is an Integral Domain

In algebra, understanding the structure of rings is fundamental. One important distinction to make is between a commutative ring that is also an integral domain. An integral domain is a commutative ring with a non-zero multiplicative identity that has no zero divisors. This article will guide through the necessary steps to prove that a commutative ring is an integral domain, even if it is known not to be a field.

Understanding the Definitions

To prove that a commutative ring R is an integral domain, we must check two key properties:

No zero divisors: For any two nonzero elements a, b ∈ R, the product ab must be nonzero. Nontriviality: The ring R must have at least two distinct elements, typically 0 and 1, where 1 ≠ 0.

Given that we know R is not a field, we can skip the requirement that every nonzero element has a multiplicative inverse. Let's delve into the proof process.

Steps to Prove R is an Integral Domain

Show that R is commutative: This property is already given as R is a commutative ring. Verify nontriviality: Ensure that R has at least two distinct elements. Confirm that 0 ≠ 1 in R since R is not the trivial ring with only one element. Prove R has no zero divisors: Take any two nonzero elements a, b ∈ R. Show that ab ≠ 0. If you can find a proof or argument that no product of two nonzero elements yields zero, then R lacks zero divisors.

Methods to Prove No Zero Divisors

Depending on the structure of the ring, you might use various techniques to demonstrate the absence of zero divisors:

Contradiction: Assume ab 0 for some nonzero a, b and derive a contradiction from the properties of the ring or the nature of the elements. Specific properties: If R has additional structures like being a subset of a certain field or having specific algebraic properties, leverage those structures to show that ab ≠ 0.

Example Considerations

If R is given as a specific ring like ? (the set of integers) or ?/n? (the set of integers modulo n, where n is not prime), you can directly analyze the elements and their products.

For ?, since every nonzero integer has a multiplicative inverse, there are no zero divisors. For ?/n?, you need to check specific values and their products modulo n.

If R is defined by some generators or relations, analyze those to demonstrate the absence of zero divisors.

Conclusion

By successfully showing that R is commutative, nontrivial, and has no zero divisors, you can conclude that R is an integral domain, even if it is not a field.