# Basic Algebra

• Ray Mines
• Fred Richman
• Wim Ruitenburg
Part of the Universitext book series (UTX)

## Abstract

A monoid is a set G together with a function ϕ from G×G to G, usually written as ϕ(a, b) = ab, and a distinguished element of G, usually denoted by 1, such that for all a, b, c in G
1. (i)

(ab)c = a(bc), (associative law)

2. (ii)

1a = a1 = a. (identity)

The function ϕ is called multiplication and the element 1 the identity. The associative law allows us to ignore parentheses in products a1a2a n . The monoid is said to be abelian, or commutative, if ab = ba for all elements a and b. In an abelian monoid the function ϕ is often called addition and written as ϕ(a, b) = a + b; the identity element is then denoted by 0. In this case we speak of an additive monoid, as opposed to a multiplicative monoid. In a multiplicative monoid we write the n-fold product aaa as a n for each positive integer n, and set a0 =1; in an additive monoid we write the n-fold sum a + a + ⋯ + a as na, and set 0a = 0.

## Keywords

Prime Ideal Commutative Ring Left Ideal Free Module Division Ring
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.