Identifying the False Statement: Integral Domains and Fields

The question asks to identify the statement that is incorrect among the given options related to the properties of integral domains and fields in abstract algebra.

Option 2: Integral Domain to Field Analysis

• Statement:
  Every integral domain is field.
• Verdict: This statement is FALSE.
• Reasoning: An integral domain is defined as a commutative ring with unity (multiplicative identity, 1) that has no zero divisors. A field is a commutative ring with unity where every non-zero element possesses a multiplicative inverse.
• Counterexample: Consider the ring of integers, denoted as $\mathbb{Z}$. This ring is commutative, has unity (1), and lacks zero divisors (if $a \cdot b = 0$ in $\mathbb{Z}$, then $a=0$ or $b=0$). Thus, $\mathbb{Z}$ is an integral domain. However, $\mathbb{Z}$ is not a field because many non-zero elements do not have multiplicative inverses within $\mathbb{Z}$. For example, the element 2 lacks a multiplicative inverse in $\mathbb{Z}$, as $1/2$ is not an integer.
• Conclusion: Since $\mathbb{Z}$ is an integral domain but not a field, the statement "Every integral domain is field" is false.

Verification of Other Options

• Option 1: Every field is an integral domain. This is TRUE. In a field, every non-zero element $a$ has a multiplicative inverse $a^{-1}$. If $a \neq 0$ and $ab=0$, multiplying by $a^{-1}$ yields $b=0$. Therefore, fields have no zero divisors.
• Option 3: Finite integral domain is field. This is TRUE. It is a well-established theorem in ring theory that any integral domain containing a finite number of elements must also be a field.
• Option 4: The ring $Z_p$ of integers modulo p is field iff p is prime. This is TRUE. The ring of integers modulo $p$, denoted $\mathbb{Z}_p$, is a field if and only if $p$ is a prime number. If $p$ is composite, $\mathbb{Z}_p$ contains zero divisors and is not a field.