A nonunital algebra is like an associative algebra but without specified identity element. The associative algebra version of nonunital ring, see there for more.
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 , simply define its product to be for all vectors . Note that since a morphism is a morphism of vector spaces such that , 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.
A definition of algebraic K-theory for nonunital rings is due to
with further developments (in KK-theory) including
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