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Axiom asserting existence of a multiplicative identity element in a ring.
The unital ring axiom asserts that a ring has an element such that
A unital ring is a ring satisfying this axiom. The presence of allows one to define units (and hence the group of units) and is typically assumed when discussing standard constructions such as polynomial rings and quotients.
A ring is a set equipped with two binary operations and such that:
These conditions are often summarized as the ring axioms. Many texts additionally impose a multiplicative identity, leading to a unital ring; commutativity of multiplication leads to a commutative ring.
Let be a unital ring. An element is a unit if there exists such that .