Ring

A ring is an algebraic structure on a set, consisting of both an Abelian group and a monoid over the set which satisfy a distributivity condition — equivalently, rings are monoids in $𝖠 𝖻$ .

That is a ring $(𝑅, +, \cdot)$ consists of an Abelian group $(𝑅, +)$ called addition and a Monoid $(𝑅, \cdot)$ called multiplication, with the extra conditions¹ #m/def/ring

left-distributivity $𝑎 \cdot (𝑏 + 𝑐) = (𝑎 \cdot 𝑏) + (𝑎 \cdot 𝑐)$
right-distributivity $(𝑏 + 𝑐) \cdot 𝑎 = (𝑏 \cdot 𝑎) + (𝑐 \cdot 𝑎)$

A ring may be generalized to a Rng (possibly lacking unity, where multiplication need only be a Semigroup), or a Rig (possibly lacking additive inverses, where addition need only be an abelian monoid), or specified to an Integral domain or Field (where both operations form abelian groups ignoring the additive identity, i.e. every element except $0$ is a unit²).

Terminology

A Subring is a subset of a ring which is itself a ring (under the same operations)

Properties

A ring has all the properties of a Rng, in addition:

$(- 1) 𝑎 = - 𝑎$
$(- 1) (- 1) = 1$

Proof of 1–2

Both ^P1 and ^P2 follow directly from Properties.

Examples

Zero ring
Polynomial ring (Gallian §16 pp. 276ff.)
$ℤ$
Adjunction of a ring

#state/tidy | #lang/en | #SemBr

2009. Algebra: Chapter 0, §III.1.1, pp. 119–120 ↩
A multiplicative unit is an element with a multiplicative inverse. A Zero-divisor can multiply a nonzero element to give zero. An element cannot be both. ↩

Ring

Terminology

Properties

Examples

Footnotes