The set \( \mathbb{Z} \) together with addition and multiplication forms a ring. The polynomials with ordinary addition and multiplication also form a ring. \( n\times n \) matrices also form a ring under addition and multiplication. Formally, a ring is a set \( R \) with two operations \( + \) and \( \times \) such that:

  1. R is an abelian group under \( + \) (that is, R fulfills all the conditions for a group G1 to G4, plus commutativity, G5).
  2. There is a multiplicative identity \( e \), such that \( a\times e=e\times a=a \). Frequently, we use \( e=1 \).
  3. Multiplication is associative.
  4. We have the distributive property of multiplication concerning addition.