# Ring

## Dependencies:

$(R, +, \circ)$ is a ring iff all of the following conditions are met (here $ab$ means $a \circ b$):

- $(R, +)$ is a commutative group (identity is denoted as 0).
- Closure of multiplication: $\forall a, b \in R, ab \in R$.
- Associativity of multiplication: $\forall a, b, c \in R, (ab)c = a(bc)$.
- Distributive property: $\forall a, b, c \in R, (a(b+c) = ab+ac \wedge (a+b)c = ac+bc)$.

A ring is said to be commutative if the $\circ$ operator is commutative.

Common examples of commutative rings are $(\mathbb{Q}, +, \times)$, $(\mathbb{R}, +, \times)$ and $(\mathbb{C}, +, \times)$ (it is trivial to prove that they are commutative rings).

## Dependency for:

- Ideal
- 0x = 0 = x0
- Field
- Zn is a ring
- (-a)b = a(-b) = -ab
- Conditions for a subset of a ring to be a subring
- Characteristic of ring equals additive order of unity
- Integral Domain
- Ring isomorphism
- Vector optional
- Matrix
- Determinant
- Polynomial

## Info:

- Depth: 1
- Number of transitive dependencies: 1