Zero divisor

In abstract algebra , a zero divisor of a ring is an element for which there is an element different from the zero element 0 , so that . ${\displaystyle R}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ab=0}$

definition

If a ring is and , a distinction is made between: ${\displaystyle R}$ ${\displaystyle a\in R}$

• Left zero divider : There is an element such that . ${\displaystyle b\in R\setminus \{0\}}$ ${\displaystyle ab=0}$
• Right zero divider : There is an element such that . ${\displaystyle b\in R\setminus \{0\}}$ ${\displaystyle ba=0}$
• (two-sided) zero divisor : is both left and right zero divisor . ${\displaystyle a}$
• Left non-zero divider : is not a left zero divider. ${\displaystyle a}$
• Right null divider : is not a right null divider. ${\displaystyle a}$
• (two-sided) non-zero divisor : is neither left nor right zero divisor, often also called a regular element . ${\displaystyle a}$

In non-commutative rings, left zero divisors do not have to be right zero divisors and vice versa, in commutative rings, on the other hand, all six terms simply coincide with zero divisors or non- zero divisors .

Some authors do not allow 0 as a zero divisor, they require it . Otherwise, left, right or two-sided zero divisors other than 0 are called real . A ring without real left and without real right zero divisors is called zero divisor -free . ${\displaystyle a\neq 0}$

A commutative ring with zero divisors and one element is called an integrity ring . ${\displaystyle 1\neq 0}$

Examples

• The ring of whole numbers does not have zero divisors , the ring (with component-wise addition and multiplication) contains, for example, the zero divisors and , because and . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} ^{2}}$ ${\displaystyle (0,1)}$ ${\displaystyle (1,0)}$ ${\displaystyle (0,1)\cdot (1,0)=(0,0)}$ ${\displaystyle (1,0)\cdot (0,1)=(0,0)}$

• Every body has zero divisors, because every element other than 0 is a unit (see below).

• The remainder class ring has zero divisors 2, 3, and 4 because it is . ${\displaystyle \mathbb {Z} /6\mathbb {Z} }$ ${\displaystyle 2\cdot 3\equiv 4\cdot 3\equiv 0\mod 6}$

• In general, for a natural number the remainder class ring is zero- divisor -free (even a field) if and only if is a prime number . ${\displaystyle n>1}$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle n}$

• For example, the ring of real 2 × 2 matrices contains the zero divisors

${\displaystyle {\begin{pmatrix}1&1\\2&2\end{pmatrix}}\quad {\text{und}}\quad {\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}}$

because

${\displaystyle {\begin{pmatrix}1&1\\2&2\end{pmatrix}}\cdot {\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.}$

• In general, in a matrix ring over a body or integrity ring, precisely the matrices whose determinant is 0 are zero divisors . Despite the lack of commutativity, there is no difference between left and right zero dividers.

properties

• In rings, an element not equal to zero is a left, right or two-sided non-zero divisor if and only if it can be shortened to the left, right or two-sided .

• Real zero divisors are not units , because if zero divisors and invertible, that is, for a suitable one , then would be . ${\displaystyle a\neq 0}$ ${\displaystyle ab=0}$ ${\displaystyle b\neq 0}$ ${\displaystyle 0=a^{-1}\cdot 0=a^{-1}ab=b}$

• In a non-commutative ring with one element ( for all ) this statement is only valid as follows: ${\displaystyle 1\cdot a=a\cdot 1=a}$ ${\displaystyle a}$

A left zero divider does not have a left inverse, but a left zero divider can have a right inverse. The same applies to right zero divisors. A double-sided zero divider therefore has no inverse.

• If there is a left zero factor, then obviously the product is also a left zero factor or zero for each product . The product , on the other hand, does not have to be a left or right zero divisor (see the example of the matrix ring in the article unit (mathematics) , whose elements and one-sided zero divisors are one-sided inverses of each other, as is the identity matrix ). ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ba}$ ${\displaystyle ab}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle AB=E}$

See also

• Topological zero divisor

Individual evidence

1. ^ BL van der Waerden: Algebra I , Springer-Verlag (1971), 8th edition, ISBN 3-540-03561-3 , §11, page 36
2. ↑ G. Fischer, R. Sacher: Introduction to Algebra , Teubner-Verlag (1978), ISBN 3-519-12053-4 , definition 1.1.7
3. ↑ Jens Carsten Jantzen: Algebra , Springer-Verlag 2013, ISBN 978-3-642-40532-7 , chap. III, §2: units, zero divisors
4. ↑ Jens Carsten Jantzen: Algebra , Springer-Verlag 2013, ISBN 978-3-642-40532-7 , chap. III, §2, example 2.4. (2)
5. ^ Kurt Meyberg: Algebra. Volume 1. Hanser, Munich et al. 1980, ISBN 3-446-13079-9 , Lemma 3.2.15