Zero Divisor — Definition, Formula & Examples
A zero divisor is a nonzero element in a ring such that for some nonzero element . In other words, two nonzero things can multiply together to give zero, which never happens with ordinary integers or real numbers.
Let be a ring. A nonzero element is called a zero divisor if there exists a nonzero element such that or . If , then is a left zero divisor; if , then is a right zero divisor. In a commutative ring, these notions coincide.
How It Works
To determine whether an element is a zero divisor, you look for a nonzero partner that produces zero under multiplication. In , for instance, , so both and are zero divisors. A ring with no zero divisors (other than itself) is called an integral domain. The integers , for example, form an integral domain because the product of two nonzero integers is always nonzero.
Worked Example
Problem: Determine whether is a zero divisor in .
Step 1: We need to find a nonzero element in such that , i.e., .
Step 2: Try : compute , and . Since in , we have found a nonzero partner.
Answer: Yes, is a zero divisor in because with .
Why It Matters
Zero divisors determine fundamental structural properties of rings. A commutative ring without zero divisors is an integral domain, which guarantees cancellation laws and enables fraction-field constructions. Recognizing zero divisors is essential in abstract algebra courses and arises in coding theory, where properties of polynomial rings over finite fields depend on the absence of zero divisors.
Common Mistakes
Mistake: Calling itself a zero divisor.
Correction: By definition, a zero divisor must be a nonzero element. The element satisfies trivially for all , so it is excluded to keep the concept meaningful.