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.
Explicitly, as an -module, with product given by
(a_1, 0) (a_2, 0) = (a_1 a_2, 0) ,
(0, r_1) (0, r_2) = (0, r_1 r_2) ,
(a,0) (0,r) = (0,r) (a,0) = (r a, 0) ,
or in general
(a_1, r_1) (a_2, r_2) = (a_1 a_2 + r_2 a_1 + r_1 a_2, r_1 r_2) .
We often write as or , which makes the above formulas obvious.