Zero divisors

 

Zero Divisors in a Ring

In ring theory, a zero divisor is an element in a ring that multiplies with another nonzero element to produce zero. Zero divisors exist in some rings but not in others, depending on their structure.

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Definition of Zero Divisors

An element a in a ring $R$ (with multiplication $*$) is called a zero divisor if there exists a nonzero element $b \in R$ such that:

$$a * b = 0 \quad \text{or} \quad b * a = 0, \quad \text{where } b \neq 0.$$

This means that multiplication in the ring does not always behave like it does in the integers or real numbers.

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Examples of Zero Divisors

1. Modulo Ring $\mathbb{Z}_6$ The ring $\mathbb{Z}_6 = \{0,1,2,3,4,5\}$ under modular arithmetic has zero divisors.

   • $2 * 3 = 6 \equiv 0 \mod 6$, so 2 and 3 are zero divisors.
   • $4 * 3 = 12 \equiv 0 \mod 6$, so 4 and 3 are zero divisors.

2. Matrix Ring $M_2(\mathbb{R})$ (Set of 2×2 Matrices) Consider the matrices:

   $$A = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}$$

   Their product is:

   $$A * B = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 \cdot 0 + 1 \cdot 1 & 1 \cdot 0 + 1 \cdot 1 \\ 0 \cdot 0 + 0 \cdot 1 & 0 \cdot 0 + 0 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$$

   Since $A * B \neq 0$, these matrices are not zero divisors. However, if we had found $A * B = 0$, then $A$ and $B$ would be zero divisors.

3. Polynomial Ring $\mathbb{Z}_6[x]$ In the ring of polynomials with coefficients in $\mathbb{Z}_6$, the polynomial

   $$(2x + 3)(3x + 2) \equiv 0 \mod 6$$

   means $2x + 3$ and $3x + 2$ are zero divisors.

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Rings Without Zero Divisors: Integral Domains

A ring without zero divisors is called an integral domain. That is, in an integral domain, if $a * b = 0$, then either $a = 0$ or $b = 0$.

Examples of Integral Domains (No Zero Divisors)

1. Integers $\mathbb{Z}$

   • The only way $a * b = 0$ is if either $a = 0$ or $b = 0$.
   • There are no zero divisors in $\mathbb{Z}$, so it is an integral domain.

2. Polynomials over Fields (e.g., $\mathbb{R}[x]$)

   • In $\mathbb{R}[x]$, if $f(x) * g(x) = 0$, then one of the polynomials must be the zero polynomial.
   • Therefore, $\mathbb{R}[x]$ is an integral domain.

3. Prime Modulo Rings $\mathbb{Z}_p$ (where $p$ is prime)

   • If $p$ is prime, $\mathbb{Z}_p$ has no zero divisors, making it an integral domain.
   • Example: $\mathbb{Z}_7 = \{0,1,2,3,4,5,6\}$ is an integral domain.

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Difference Between Zero Divisors and Nonzero Elements

Element	Zero Divisor?	Example in $\mathbb{Z}_6$
0	Trivially yes	$0 * 1 = 0$
1	No	$1 * x = x$ for all $x$
2	Yes	$2 * 3 = 0$ mod 6
3	Yes	$3 * 2 = 0$ mod 6
4	Yes	$4 * 3 = 0$ mod 6
5	No	No nonzero element multiplies with 5 to give 0

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Key Takeaways

1. Zero divisors are elements in a ring that can multiply with a nonzero element to produce zero.
2. Integral domains have no zero divisors—they generalize the integers.
3. Fields are special cases of integral domains where every nonzero element has a multiplicative inverse.
4. Noncommutative rings, such as matrix rings, can have zero divisors.