Rings and domains · Definition

Zero divisor

modern algebrarings

In the integers a zero product forces a zero factor; in other rings that can fail, and zero divisors measure how much.

Definition

A zero divisor of a ring \(R\) is an element \(a\neq0\) for which there is another element \(b\neq0\) of \(R\) whose product with \(a\) is zero.

\[ab=0,\quad a\neq0,\ b\neq0\]

Zero divisors appear when the product of two nonzero elements vanishes. In \(\mathbb{Z}_6\) we have \(2\cdot3=0\), so 2 and 3 are zero divisors; in \(\mathbb{Z}_7\) there are none. In general, in \(\mathbb{Z}_n\) an element \(a\neq0\) is a zero divisor exactly when \(\gcd(a,n)>1\), and a unit when \(\gcd(a,n)=1\): every nonzero element is one or the other.

A unit is never a zero divisor, since if \(uv=0\) multiplying by \(u^{-1}\) gives \(v=0\). In a finite commutative ring with identity the converse holds: every nonzero element is a unit or a zero divisor. Beyond finite rings it can fail: in \(\mathbb{Z}\) the number 2 is neither a unit nor a zero divisor.

The practical effect is cancellation. If \(a\) is not a zero divisor and \(ab=ac\), then \(a(b-c)=0\) forces \(b=c\); if it is one, cancellation is lost: in \(\mathbb{Z}_8\), \(4\cdot1=4\cdot3\) without \(1=3\). An integral domain is precisely a commutative ring with identity and no zero divisors, the natural setting for factoring and solving polynomial equations.

Two typical mistakes: zero is not called a zero divisor by convention (the definition requires \(a\neq0\)), and zero divisors do not require noncommutativity: \(\mathbb{Z}_6\) already has them. In \(2\times2\) matrices, products like \(E_{11}E_{22}=0\) produce them easily.

Think about it

Why can a field not have zero divisors?

Because every nonzero element is a unit, and a unit is never a zero divisor.

If \(ab=0\) in \(\mathbb{Z}_n\) with \(n=pq\) (distinct primes), which nonzero pair shows it?

The pair \(a=p\), \(b=q\): both are nonzero and their product is \(n\equiv0\).

Exercises 10

  1. In \(\mathbb{Z}_{12}\), which of these elements is a zero divisor?
  2. How many zero divisors does \(\mathbb{Z}_{10}\) have?
  3. Which of these rings has no zero divisors?
  4. In \(\mathbb{Z}_{15}\), which \(b\neq0\) satisfies \(6b=0\)?
  5. In the ring of \(2\times2\) real matrices, which of these matrices is a zero divisor?
  6. Which statement is true in a ring \(R\) with identity, \(R\neq0\)?
  7. In \(\mathbb{Z}\times\mathbb{Z}\), which of these elements is a zero divisor?
  8. In \(\mathbb{Z}_{8}\), which pair with \(x\neq y\) satisfies \(4x=4y\)?
  9. How many zero divisors does \(\mathbb{Z}_{21}\) have?
  10. If \(a\neq0\) satisfies \(a^2=0\) in a ring, what is \(a\)?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §7.1
  • Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 13
  • I. N. Herstein, Abstract Algebra, 3rd edition, Wiley, 1996. §3.2
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §19