Algebraic structure - Simple English Wikipedia, the free encyclopedia

# Algebraic structure

### From Simple English Wikipedia, the free encyclopedia

In mathematics an algebraic structure is a set with one, two or more binary operations on it.

With one binary operation these are the basic ones

- Semigroup

    It is a set with an operation which is associative

- Monoid

    It is a semigroup with neutral element

- Group

    It is a monoid where each element has its inverse

- Commutative Group

    It is group with a commutative operation

With two binary operations are:

- Ring

    The operation are called the sum and the product genericaly. With the sum forms a commutative group, and with the product it is a semigroup. There is distributivity between the product and the sum

- Field

    It is a ring where the product is commutative. There is distributivity and all elements different from the sum's neutral element have inverse

Examples are

- The natural numbers with the sum is a semigroup and also a monoid. It is not a group

- The integer numbers with the sum is group and also a commutative group

- The integer numbers with the sum and multiplication is a ring, but not a field

- The rational numbers, the real numbers and the complex numbers with the ordinary sum and ordinary multiplication are fields.

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Category: Mathematics

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