The unitalization of a non-unital algebra is a unital algebra with a unit freely adjoined.
For (non)associative algebras
For a commutative ring write for the category of nonassociative algebras with unit over and unit-preserving homomorphisms, and write for nonunital nonassociative -algebras. Note that is a subcategory of , as we use both ‘non-unital’ and ‘non-associative’ in accordance with the red herring principle.
The inclusion functor has a left adjoint . For we say is the unitalization of .
Explicitly, as an -module, with product given by
or in general
We often write as or , which makes the above formulas obvious.
If is an associative algebra, then will also be associative; if is a commutative algebra, then will also be commutative.
Revised on January 14, 2012 07:08:55