Definition
Category of unital associative algebras
Unital associative algebras over a field and unit-preserving linear algebra maps.
Fix a field . The category has unital associative -algebras as objects. A morphism is -linear, multiplicative, and unit-preserving:
Identities and composition are ordinary map identities and composition. Equivalently the maps preserve the ring operations and commute with the specified central scalar maps from .
Comparison with arbitrary maps
For a nonzero target, the zero linear map is not a morphism here. A bijective multiplicative linear map between unital algebras is automatically unital, since its value on the source unit acts as a unit on the entire target. Hence adding the unit condition changes general Hom-sets and endomorphism monoids, but not automorphism groups of unital objects.
The catalogue's -algebra versus -algebra comparisons use this unit-preserving convention unless the arbitrary-map category is named explicitly.