Rings

Entry conditions

You should be comfortable with groups and monoids.

Definitions

A ring $(R, +, \cdot, 0, 1)$ is a set $R$ with two binary operations satisfying:

1. $(R, +, 0)$ is an abelian group.
2. $(R, \cdot, 1)$ is a monoid.
3. Multiplication distributes over addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c) \cdot a = b \cdot a + c \cdot a$.

If $\cdot$ is also commutative, the ring is commutative.

Vocabulary (plain language)

• Ring: a set with addition and multiplication where addition forms a group and multiplication distributes over addition.
• Commutative ring: a ring where $a \cdot b = b \cdot a$.
• Unit: an element with a multiplicative inverse.

Symbols used

• $(R, +, \cdot)$: a ring with addition and multiplication.
• $0$: the additive identity.
• $1$: the multiplicative identity.

Intuition

A ring is what you get when you take integers as a model and ask: what properties do addition and multiplication really need? The integers $\mathbb{Z}$ satisfy all the ring axioms. So do polynomials, matrices, and many other objects.

The key idea is that addition is fully reversible (it forms a group) while multiplication need not be — you cannot always divide in a ring. Structures where you can always divide (by nonzero elements) are fields.

Worked example

The integers $(\mathbb{Z}, +, \cdot, 0, 1)$ form a commutative ring. Addition gives an abelian group. Multiplication is associative with identity $1$ and distributes over addition. But most integers have no multiplicative inverse in $\mathbb{Z}$ — for instance, there is no integer $n$ with $2n = 1$.

Common mistakes

• Forgetting that ring multiplication need not be commutative. Matrix rings are non-commutative.
• Assuming every nonzero element has a multiplicative inverse. That is a field, not just a ring.