Solution

The correct answer is Every integral domain is field.

Key Points

• An integral domain is a commutative ring with unity (1 ≠ 0) that has no zero divisors. This means that if a and b are any two elements of the domain, and a*b = 0, then either a = 0 or b = 0.
• A field is a commutative ring with unity in which every non-zero element has a multiplicative inverse.
• While every field is an integral domain (because fields have no zero divisors), not every integral domain is a field.
• The distinction lies in the requirement for multiplicative inverses. Integral domains do not guarantee that every non-zero element has an inverse, which is a defining property of fields.

Additional Information

• Finite Integral Domains: A finite integral domain is always a field. This is because the absence of zero divisors in a finite set ensures that every non-zero element has a multiplicative inverse.
• Examples:
  • The set of integers modulo a prime p, denoted as Z_p, forms a field because every non-zero element has a multiplicative inverse when p is a prime.
  • The set of integers Z is an integral domain, but it is not a field because not all non-zero integers have multiplicative inverses within Z.
• Key Misunderstanding: The statement "Every integral domain is a field" is false because the property of having multiplicative inverses is not guaranteed in integral domains.