Definition·D061
Ring
A set with an addition making it an abelian group, and an associative multiplication that distributes over addition.
A ring is a triple
where
is an abelian group (write
for its identity,
for the inverse of
) and
is a binary operation on
such that, for all
:
In words
A ring is a set with an addition under which it is an abelian group, together with a multiplication that is associative and spreads over addition on both sides.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
If moreover there is a unity
with
for every
, the ring is called unital (or with unity); if
is commutative the ring is called commutative. The integers are the prototypical example: a commutative ring with unity. Elementary consequences of the axioms alone (e.g.
) are collected in L45.