Group, Abelian Group, Ring

Continuing the magma hierarchy upward.

Group $\langle G, \circ, e, \cdot^{-1} \rangle$ (Def 1.9, notes p.127):
A monoid + an inverse for every element:

So a group has: closure, associativity, identity, AND inverses. Example: $\langle \mathbb{Z}, +, 0, -\rangle$ — the integers under addition.

Abelian group $\langle G, \circ, e, \cdot^{-1}, c \rangle$ (Def 1.10, notes p.128):
A group + commutativity:

Ring $\langle R, +, \cdot, 0, -, 1 \rangle$ (Def 1.12, notes p.129; slides p.350 Example 1.12):
A ring has two binary operations, $+$ and $\cdot$, satisfying:
- $\langle R, +, 0, -\rangle$ is a (abelian) group,
- $\langle R, \cdot, 1\rangle$ is a monoid,
- Distributivity: $\forall a, b, c \in R.\ a \cdot (b + c) = a \cdot b + a \cdot c$.

So a ring glues a group and a monoid together and demands that multiplication distribute over addition. Example: $\langle \mathbb{Z}, +, \cdot, 0, -, 1\rangle$.

Sanity check — Example 1.9 in the slides: $\langle \mathbb{Z}, \cdot, 1\rangle$ is a monoid, but not a group, because integers are not closed under division. You'd need the rationals $\mathbb{Q}$ for a multiplicative group.