# Neutral element

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.

## definition

Be a magma (a quantity with a two-digit link ). Then an element is called${\ displaystyle (S, *)}$ ${\ displaystyle e \ in S}$ • left neutral , if is for everyone ,${\ displaystyle e * a = a}$ ${\ displaystyle a \ in S}$ • legally neutral , if is for everyone ,${\ displaystyle a * e = a}$ ${\ displaystyle a \ in S}$ • neutral if left neutral and right neutral.${\ displaystyle e}$ 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 . ${\ displaystyle S}$ ${\ displaystyle S}$ ${\ displaystyle S}$ ${\ displaystyle S}$ 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 . ${\ displaystyle *}$ ${\ displaystyle \ cdot}$ ${\ displaystyle 1}$ ${\ displaystyle \ cdot}$ 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. ${\ displaystyle *}$ ${\ displaystyle +}$ ${\ displaystyle 0}$ ## Examples

• In the real numbers is ( zero ) the neutral element of addition and ( one ) the neutral element of multiplication , as and for any real number .${\ displaystyle 0}$ ${\ displaystyle 1}$ ${\ displaystyle 0 + x = x + 0 = x}$ ${\ displaystyle 1 \ times x = x \ times 1 = x}$ ${\ displaystyle x}$ • In the ring of - matrices over a body , the zero matrix is the neutral element of the matrix addition and the identity matrix is the neutral element of the matrix multiplication .${\ displaystyle n \ times n}$ • In a function space , the null function is the neutral element of addition and the one function is the neutral element of multiplication.
• In the case of vectors , the zero vector is the neutral element of the vector addition .
• In a formal language , the empty word is the neutral element of the concatenation of words.

## properties

• 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 .${\ displaystyle S}$ ${\ displaystyle S}$ ${\ displaystyle a * e = a}$ ${\ displaystyle f * a = a}$ ${\ displaystyle a \ in S}$ ${\ displaystyle f = f * e = e}$ • 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.${\ displaystyle a * b: = b}$ • This can also occur when multiplying in rings. One example is the partial ring
${\ displaystyle R = \ left \ {\ left. {\ begin {pmatrix} a & b \\ 0 & 0 \ end {pmatrix}} \ right | a, b \ in K \ right \}}$ 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 ${\ displaystyle K}$ ${\ displaystyle R}$ ${\ displaystyle {\ begin {pmatrix} 1 & x \\ 0 & 0 \ end {pmatrix}}}$ with . According to what has been said above, the multiplication in cannot have any right-neutral elements.${\ displaystyle x \ in K}$ ${\ displaystyle R}$ 