Rings

A ring is a set together with two binary operations $+$ and $\cdot$, which we will call addition and multiplication, such that the following axioms are satisfied:

  1. $\langle R, + \rangle$ is an abelian group.

  2. Multiplication is associtative.

  3. For all $a,b,c \in R$, the left distributive law and the right distribute law hold, i.e.

\[a \cdot (b + c) = (a \cdot b) + (a \cdot c), \quad (a + b) \cdot c = (a \cdot c) + (b \cdot c).\]

For example, the integers, rationals, reals and complex numbers are all rings with the usual addition and multiplication.

A ring homomorphism $\phi : R \to R’$ must satisfy the following two properties:

  1. $\phi{(a+b)} = \phi{(a)} + \phi{(b)}.$

  2. $\phi{(ab)} = \phi{(a)}\phi{(b)}.$

A ring doesn’t have to have a multiplicative identity element, but it can. If it has one, it’s denoted $1$ and for all $a$ in $R$ satisfies $a1 = 1a = a$. The element $1$ is also called unity.

A ring in which multiplication is commutative is called a commutative ring. A ring that has a multiplicative identity element is called a ring with unity.

For some element $a$ in a ring with unity $R$ where $1 \neq 0$, if $a^{-1} \in R$ such that $aa^{-1} = a^{-1}a = 1$, $a^{-1}$ is said to be the multiplicative inverse of $a$ and $a$ is said to be a unit in $R$.

Fields

Let $R$ be a ring with unity. If every nonzero element of $R$ is a unit (has a multiplicative inverse), then $R$ is called a division ring. A commutative division ring is called a field.

For example, the integers are not a field, but the rationals, the reals, and the complex numbers are all fields.