Ring

Dependencies:

1. Group

$(R, +, \circ)$ 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:

1. Ideal
2. 0x = 0 = x0
3. Field
4. Zn is a ring
5. (-a)b = a(-b) = -ab
6. Conditions for a subset of a ring to be a subring
7. Characteristic of ring equals additive order of unity
8. Integral Domain
9. Ring isomorphism
10. Vector optional
11. Matrix
12. Determinant
13. Polynomial

Info:

• Depth: 1
• Number of transitive dependencies: 1

Transitive dependencies:

1. Group