nLab nonunital algebra

Idea

A nonunital algebra is like an associative algebra but without specified identity element. The associative algebra version of nonunital ring, see there for more.

Examples

The zero algebra is an example of an algebra that is nonunital as soon as the underlying vector space has dimension greater than 0. Given a vector space $V$, simply define its product to be $\mu: v\otimes w\mapsto 0$ for all vectors $v,w\in V$. Note that since a morphism $f:(A,\mu)\to (A',\mu')$ is a morphism of vector spaces $f:A\to A'$ such that $f(\mu(v\otimes w))=\mu'(f(v)\otimes f(w))$, a morphism between zero algebras is the same as linear maps of the underlying vector spaces. Thus, the zero product describes a full embedding of the category of vector spaces into the category of (nonunital) algebras.

Related concepts

• unitisation of C*-algebras.

• noncommutative algebra, nonassociative algebra

References

A definition of algebraic K-theory for nonunital rings is due to

• Daniel Quillen, $K_0$ for nonunital rings and Morita invariance, J. Reine Angew. Math., 472:197-217, 1996.

with further developments (in KK-theory) including

• Snigdhayan Mahanta, Higher nonunital Quillen K’-theory, KK-dualities and applications to topological T-dualities, J. Geom. Phys., 61 (5), 875-889, 2011. (pdf)