nLab
associative unital algebra

Definitions

An associative unital algebra over a given field (or even commutative ring) $k$ can be defined as (equivalently)

• a monoid internal to Vect;

• a one-object category enriched over (Vect,$\otimes$);

• an algebroid with one object;

• a vector space $V$ equipped with linear maps $p:V\otimes V\to V$ and $i:k\to V$ satisfying the associative and unit laws;

• a ring under the ground field $k$.

If there is no danger for confusion, one often says simply ‘associative algebra’, or even only ‘algebra’.

More generally, a (merely) associative algebra need not have $i:k\to V$; that is, it is a semigroup instead of a monoid.

Less generally, a commutative algebra (where associative and unital are usually assumed) is an abelian monoid in $\mathrm{Vect}$.

Variants

• A cosimplicial algebra is a cosimplicial object in the category of algebras.

• A dg-algebra is a monoid not in Vect but in the category of (co)chain complexes.

• A smooth algebra is an associative $ℝ$-algebra that has not only the usual binary product induced from the product $ℝ\times ℝ\to ℝ$, but has a $n$-ary product operation for every smooth function ${ℝ}^n\to ℝ$.

  This may be understood as a special case of an algebra over a Lawvere theory, here the Lawvere theory CartSp.