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
That is a ring
- left-distributivity
- right-distributivity
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
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:
Proof of 1–2
Both ^P1 and ^P2 follow directly from Properties.
Examples
- Zero ring
- Polynomial ring (Gallian §16 pp. 276ff.)
- Adjunction of a ring
#state/tidy | #lang/en | #SemBr
Footnotes
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2009. Algebra: Chapter 0, §III.1.1, pp. 119–120 ↩
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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. ↩