Phy622 Mathematical Methods 3

Rings

$R = ({S}, *, +)$

where,

1. $({S}, +)$ is an Abelian [Group]

2. $*$ is binary note 2, not necessarily commutative, with a multiplicative identity note 1. $({S}, +)$ is a Monoid

3. Multiplication is distributive over addition.

NOTE

1. Some definitions do not require multiplicative identity. Generally these are called rngs
2. Some definitions do not require associativity of $*$.

Special Rings

• Division Ring - multiplicative inverse exists
• Lie Ring
  • $x^*y = -y^*x$
  • Jacobi identity - $x(yz) + z(xy) + y(zx) = 0$