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Math and science::Algebra::Aluffi

Rings. Additional structure.

As structure is added to rings, a number of useful concepts are graced with definitions.

  • Commutative ring: a ring whose multiplicative operation is commutative.
  • Division ring: a ring where all nonzero elements have two-sided multiplicative inverses.
  • Integral domain: a ring whose multiplicative operation is commutative and no nonzero element is a zero divisor.
  • Field: satisfies both 1) and 2) above.

The zero ring is specifically excluded from being considered to be either an integral domain or a field, despite otherwise meeting the requirements.


Vanilla ring

For the vanilla ring \( (R, +, \cdot ) \), the second operation \( ( R, \cdot) \) doesn't need to be commutative, nor do elements need to have inverses with respect to this operation.

Adding commutativity:

Commutative rings

Rings where the multiplicative operation is commutative are called commutative rings.

Adding inverses (two-sided):

Division rings

Rings where every element except the zero element has a two-sided inverse with respect to the multiplicative operation are called division rings.

The zero element (identity for the group operation) cannot have an inverse. One justification is to remember that, as endomorphisms of an abelian group, the zero element is the endomorphism mapping everything to the identity, and composition with this function is still the endomorphism mapping everything to the identity.

Commutativity without zero-divisors:

Integral domain

A commutative ring \( (R, +, \cdot) \) such that:

\[ \forall a, b \in R, \; ab = 0 \implies (a = 0 \lor b = 0) \]

is called an integral domain.

The zero ring is excluded even though it meets the above requirement.

The cyclic groups \( \mathbb{Z}/n\mathbb{Z} \) form commutative groups when multiplication is added, but they only form integral domains when \( n \) is a prime.

Commutativity and inverses:

Fields

A field is a non-zero commutative division ring.

In other words, a field is a non-zero commutative ring where all elements have a two-sided inverse.

Fields are integral domains

All fields are integral domains, but there are integral domains which are not fields.

If a non-zero element has a multiplicative inverse, then it cannot be a zero divisor. And so, fields have no zero-divisors, making them integral domains.

Finite integral domains are fields

By first assuming a ring is finite, we can then promote the statement in the previous section:

\[ \text{Let } R \text{ be a finite ring, then } \; R \text{ is an integral domain } \iff R \text{ is a field.} \]