Rings

Definition: Ring @ring 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: $\langle R, + \rangle$ is an abelian group. Multiplication is associative. 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.

Definition: Ring Homomorphism @ring-homomorphism A ring homomorphism $\phi : R \to R'$ must satisfy the following two properties: $\phi{(a+b)} = \phi{(a)} + \phi{(b)}.$ $\phi{(ab)} = \phi{(a)}\phi{(b)}.$

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 $a1 = 1a = a$.

Definition: unity @unity The element $1$ is also called unity .

Definition: Commutative Ring @commutative-ring A ring in which multiplication is commutative is called a commutative ring .

Definition: Ring with Unity @ring-with-unity A ring that has a multiplicative identity element is called a ring with unity.

Definition: Multiplicative Inverse @multiplicative-inverse 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 $a.$

Definition: Unit @unit If $a$ has a multiplicative inverse in $R,$ $a$ is said to be a unit in $R$. Referenced by (2 direct) Direct references: Surface Integral over Vector Field Stoke's Theorem

Fields

Definition: Division Ring @division-ring 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.

Definition: Field @field A commutative division ring is called a field . Referenced by (2 direct, 33 transitive) Direct references: Vector Space Scalar Transitive (depth 1): remark-32 Vector Multiplication by a Scalar note-3 Hessian Matrix Vector Transitive (depth 2): remark-9 remark-16 incompressible Bound vector Direction invariance-of-curl Jacobian Matrix Cross Product Unit Vector note-5 Surface Normal Vector theorem-19 remark-45 Free vector Transitive (depth 3): Vector Equality proof-of-theorem-19 remark-46 Normal Derivative Directional Derivative Zero Vector remark-30 directional-derivative-is-inner-product-of-vector-and-grad remark-23 Tangent Vector gradient-as-surface-normal-vector proof-of-gradient-as-surface-normal-vector Transitive (depth 4): Stoke's Theorem grad-div-curl-related

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