Explore Examples of Integral Domains That Are Not Fields or Rings
Understanding Integral Domains That Are Not Fields or Rings
Integral domains play a crucial role in abstract algebra. A commutative ring with a multiplicative identity that has no zero divisors is known as an integral domain. However, it's important to distinguish between rings and fields. While all fields are integral domains, not all integral domains are fields. This distinction highlights the need for a multiplicative inverse for every non-zero element in a field, a property that integral domains do not necessarily possess.
The Ring of Integers as an Example
A classic example of an integral domain that is not a field is the (mathbb{Z}), the set of integers. Let's delve into the properties that make (mathbb{Z}) an integral domain but not a field:
Ring Properties
Closure: The sum and product of any two integers are integers. Associativity: Addition and multiplication are associative operations. Commutativity: Both addition and multiplication are commutative. Identity Element: The integer 0 serves as the additive identity and the integer 1 acts as the multiplicative identity. Additive Inverses: For every integer (a), there exists an integer (-a) such that (a (-a) 0). No Zero Divisors: If (a, b in mathbb{Z}) and (ab 0), then either (a 0) or (b 0).These properties confirm that (mathbb{Z}) is an integral domain. However, (mathbb{Z}) is not a field because not all non-zero elements have multiplicative inverses within (mathbb{Z}). For instance, the integer 2 does not have an inverse in (mathbb{Z}) since (frac{1}{2}) is not an integer.
Exploring Non-Commutative Rings
Integral domains can also exist in non-commutative rings. To define an integral domain in a non-commutative setting, the key requirements are compatible addition and multiplication, formally expressed as the axioms of a ring. Here, we explore examples of non-commutative and possibly non-unital rings:
Skew-Fields and Quaternions
If commutativity is not a requirement, skew-fields such as the skew field of quaternions satisfy the properties of an integral domain. The skew-field of quaternions, denoted as (mathbb{H}), is a non-commutative extension of the complex numbers.
Principal Ideal Domains and Polynomial Rings
Another example is the ring of polynomials over a principal ideal domain (PID), such as (A[X]). This ring is an integral domain and any proper non-zero ideal within it serves as a non-commutative, non-unital example. For instance, if (A) is a PID, the set of polynomials with coefficients from (A) but not including zero-degree polynomials forms a non-unital ring that is still an integral domain.
Non-Ring Structures
For algebraic structures that are not strictly rings, consider polynomials with natural or positive coefficients. Although these structures do not meet all the axioms of a ring, they still exhibit properties consistent with integral domains. Essentially, you can modify existing rings to form structures that maintain the integral domain property without necessarily forming a ring.
Conclusion
The understanding of integral domains that are not fields or rings is essential in algebraic structures. Whether in commutative or non-commutative settings, such examples help us explore the vast landscape of algebraic structures and their unique properties.
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