Zero divisor

In abstract algebra, a zero divisor of a ring is the zero element 0 miscellaneous item for which there is a zero element 0 of the element that is different, so.

Definition

Is a ring and, then, a distinction between:

  • Left zero divisor: There is an element such that.
  • Right zero divisor: There is an element such that.
  • (two-sided ) zero divisor: is both left-and right zero divisor.
  • Links Non- zero divisor: No links zero divisor.
  • Not right zero divisor: is not a right zero divisor.
  • (two-sided ) non- zero divisor: neither left nor right zero divisor, often called regular element.

In non-commutative rings left zero divisor need not be right zero divisor, and vice versa, in commutative rings, however, includes all six terms simply to zero divisors and non- zero divisor together.

Some authors also allow for the 0 and zero divisors, that is, they waive the condition. Then 0 is always a zero divisor and is called real different from 0 left, right or two-sided zero divisor. A ring with no real links and no real right zero divisor is called zero divisors.

A zero-divisor free and commutative ring with unit element is called integrity ring.

Examples

The ring of integers is zero divisors, ring ( with componentwise addition and multiplication ) for example, the zero divisor and, because.

Every body is zero divisors, since each different from 0 is a unit ( see below).

The residue class ring has zero divisors 2, 3 and 4, because.

General is a natural number of the residue class ring if and only zero divisors ( even a body ) when a prime number.

The ring of real 2x2 matrices, for example, contains the zero divisor

Because

Generally, in a matrix ring over a field or integral domain exactly the zero divisor matrices, which are not the zero matrix and its determinant is 0. ( There is, despite the lack commutativity no difference between left and right zero divisors ).

Properties

In wrestling is a nonzero element left, right or two-sided non- zero divisor if it is exactly then left, right or both sides can be shortened.

Zero divisors are no units, because would be inverted and then you would.

In a non-commutative ring with unit element (for all ), this statement applies only to: A left zero divisor has no left inverse. However, a left zero divisor can have a right inverse. The same is true for right zero divisor. ( A two-sided zero divisor has therefore also no inverse. )

If left zero divisor, then obviously for each product is also a left divisor of zero or equal to zero. The product must but no left or right zero divisor (see the example given in the article of the matrix ring unit ( Mathematics ) whose elements and one-sided zero divisors are each one-sided inverses of each other, as is the identity matrix ).

Unit (ring theory)
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