Definition#

A ring is a set \(R\) together with two binary operations, written as addition and multiplication, such that:

  1. \(R\) is an Abelian group under addition.
  2. if \(a,b \in R\), then \(ab \in R\) (closure under multiplication).
  3. if \(a,b,c \in R\), then \((ab)c = a(bc)\) (associativity of multiplication).
  4. if \(a,b,c \in R\), then \(a(b+c) = ab+bc\) (distributive law).
  5. if \(a,b,c \in R\), then \((a+b)c = ac + bc\) (distributive law).

Example#

See this example.

Source#

Gregory T. Lee. Abstract Algebra. 1st ed. Springer Cham, 2018. pp. 135.