Unit (ring theory)

In mathematics we mean by a unit in a unitary ring ( ring 1) every two-sided divider of 1 ( the neutral element of the multiplication ).

So if there is out there with so and both units. In this case, means for inverse element, and to the inverse.

The quantity of units

Is the multiplication of a group, called the group of unit, less commonly, the notation is used. Note that no particular follows from the definition of the unit group that is commutative.

A ring member that is not a unit, ie non-unitary.

Examples

The noncommutative case

Is not commutative, unitary ring R, then one needs terms for one-sided units.

• An element A, which satisfies the condition as a = 1 for element B is left unit.
• An element a, that satisfies the condition ba = 1 for an item b, ie legal entity.
• An element a is called a unit if there are elements b and c are with ab = 1 and ca = 1

An element is then exactly one unit if it is a left unit and a right unit at the same time. In a commutative ring, the three concepts are the same. remains in the non - commutative case, a ( two-sided ) unit.

If a is a unit, then follows that the one-sided inverse of b and c are uniquely determined and the same, the inverse of a is thus uniquely determined and is usually denoted by.

Example

There are the following ring R in which there is a left unit A, which is not a legal entity and a legal entity B, which is not a left unit. In addition, A and B are still one-sided zero divisor.

R consists of all matrices of size " countable -by- countable " with components in the real numbers, which are only a finite number of non- zeros in each row and in each column ( total may thereby infinitely many non- zeros to be included ). R is a ring with the common matrix multiplication and matrix addition. The unit matrix E has only ones on the main diagonal and zeros elsewhere, it is the unit element of R ( the neutral element of multiplication).

A is the matrix R, the upper side in the first diagonal is only ones and zeros otherwise:

B is the transpose of A, that is the matrix below the diagonal is only ones in the first diagonal and zeros otherwise.

It is AB = E, ie, A is a left unit and B is a legal entity. For each element C of R but the product has CA in the first column of all zeros, and the product BC in the first row of all zeros. Thus, A and B be any legal entity not link unit. With the matrix D, which contains only the component D 1, 1 a one and zeros otherwise, AD = 0, and DB = 0, that A is a left- zero divisor B and a right divider zero.

A functional analytic variant of this example is the unilateral shift operator.

• Algebra