Neutral element

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A neutral element is a special element of an algebraic structure . It is characterized by the fact that each element is mapped to itself through the link with the neutral element.


Be a magma (a quantity with a two-digit link ). Then an element is called

  • left neutral , if is for everyone ,
  • legally neutral , if is for everyone ,
  • neutral if left neutral and right neutral.

If the connection is commutative , then the three terms match. But if it is not commutative, then there can be a right-neutral element that is not left-neutral, or a left-neutral element that is not right-neutral.

A semigroup with a neutral element is called a monoid . If every element in also has an inverse element in , then is a group .

The symbol is often used for the link ; one then speaks of a multiplicatively written semigroup. A neutral element is then called a single element and is symbolized by. As is common with ordinary multiplication , the painting point can be left out in many situations .

A semigroup can also be noted additively by using the symbol for the link . A neutral element is then called a null element and is symbolized by.



  • If a semigroup has both right-neutral and left-neutral elements, then all of these elements match and have exactly one neutral element. Because is and for everyone , then is .
  • The neutral element of a monoid is clearly determined.
  • But if a semigroup has no right-neutral element, then it can have several left-neutral elements. The simplest example is any set with at least two elements with the link . Each element is left neutral, but none is right neutral. Similarly, there are also semigroups with right-neutral, but without left-neutral elements.
  • This can also occur when multiplying in rings. One example is the partial ring
of the 2-by-2 matrices over any body . It is easy to calculate that is a non-commutative ring. Exactly the elements are left-neutral with regard to the multiplication
with . According to what has been said above, the multiplication in cannot have any right-neutral elements.

See also

Individual evidence

  1. ^ Siegfried Bosch : Algebra. 7th, revised edition. Springer, Berlin et al. 2009, ISBN 978-3-540-92811-9 , p. 2.