A ring is a set $(R, +, \cdot)$ with operations +, ⋅ such that

1. $(R, +)$ Abelian Group
2. $(R, \cdot)$ Monoid
3. ${\bf{Distributivity}}:\ \forall x,y,z\in R: x\cdot(y+z) = x\cdot y+x\cdot z \\ (x+y)\cdot z = x\cdot z + y\cdot z$

A ring is called commutative if $\forall x,y\in R: x\cdot y = y \cdot x$, $(R,\cdot)$ is a commutative monoid

• Matrix Multiplication / Addition is a non Communative Ring
• Continuous Functions form a communative ring
• Function Composition is not a ring

Subring Test

A subset $S \subseteq R$ is a subring if it is closed under $+,\cdot$ and

• $(S, +)$ is a subgroup of $(R,+)$
  • S is closed under subtraction $(x,y\in S \implies x-y\in S)$
• $(S, \cdot)$ is a submonoid of $(R,\cdot)$
  • S is closed under multiplication $(x,y\in S \implies x\cdot y \in S)$
• $1 \in S$

Properties inherited to subrings

1. Commutativity
2. Domain (no zero-divisors) Being a field / division ring is not inherited

Remark: We also know $0 \in S$. Since $1 \in S$, so $-1 \in S$

Example:

1. $\Z \sub \mathbb{Q} \sub \R \sub \Complex$ are subrings!

2. $Z_n = \{0,1,\cdots, n-1\} \sub \Z=\{\cdots,\ -1,\ 0,\ 1, \cdots \}$

   $0-1 = -1 \notin \Z_n$ not a subring