Associative ring

Definition

A ring is a set $R$ equipped with the following operations:

• An infix binary operation $+$, called addition.
• A prefix unary operation $-$, called the negative.
• A constant element $0$, called zero.
• A binary operation $*$, called the multiplication.

satisfying the following compatibility conditions:

• $R$ forms an abelian group with group operation $+$, inverse operation $-$, and identity element $0$.
• $R$ satisfies the two distributivity laws:
  • $a*(b+c)=(a*b)+(a*c)\mathrm{\forall }a,b,c\in R$
  • $(a+b)*c=(a*c)+(b*c)\mathrm{\forall }a,b,c\in R$
• Associativity: $a*(b*c)=(a*b)*c\mathrm{\forall }a,b,c\in R$

However, in many contexts, it is useful to study the situation where $*$ is possibly non-associative, i.e., we want to remove the last condition from the definition. We use the term non-associative ring for a ring that is not necessarily associative. Note that associative rings are non-associative rings by this definition.

A unital ring or unitary ring is a ring with an identity for multiplication, denoted $1$. A commutative ring is a ring where the multiplication is commutative. A commutative unital ring is a ring where the multiplication is both commutative and has a unit.