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unitalization

Idea

The unitalization of a non-unital algebra is a unital algebra with a unit freely adjoined.

Definition

For (non)associative algebras

For $R$ a commutative ring write $R{\mathrm{Alg}}_{\mathrm{u}}$ for the category of nonassociative algebras with unit over $R$ and unit-preserving homomorphisms, and write $R{\mathrm{Alg}}_{\mathrm{nu}}$ for nonunital nonassociative $R$-algebras. Note that $R{\mathrm{Alg}}_{\mathrm{u}}$ is a subcategory of $R{\mathrm{Alg}}_{\mathrm{nu}}$, as we use both ‘non-unital’ and ‘non-associative’ in accordance with the red herring principle.

The inclusion functor $U:R{\mathrm{Alg}}_{\mathrm{u}}\to R{\mathrm{Alg}}_{\mathrm{nu}}$ has a left adjoint $(-{)}^{+}:R{\mathrm{Alg}}_{\mathrm{u}}\to R\mathrm{Alg}\mathrm{u}$. For $A\in R{\mathrm{Alg}}_{\mathrm{nu}}$ we say $A^{+}$ is the unitalization of $A$.

Explicitly, $A^{+}=A\oplus R$ as an $R$-module, with product given by

$$(a_1,0)(a_2,0)=(a_1{a}_2,0),$$(a_1, 0) (a_2, 0) = (a_1 a_2, 0) ,

$$(0,r_1)(0,r_2)=(0,r_1{r}_2),$$(0, r_1) (0, r_2) = (0, r_1 r_2) ,

$$(a,0)(0,r)=(0,r)(a,0)=(ra,0),$$(a,0) (0,r) = (0,r) (a,0) = (r a, 0) ,

or in general

$$(a_1,r_1)(a_2,r_2)=(a_1{a}_2+r_2{a}_1+r_1{a}_2,r_1{r}_2).$$(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 $(a,r)$ as $a+r$ or $a\oplus r$, which makes the above formulas obvious.

If $A$ is an associative algebra, then $A^{+}$ will also be associative; if $A$ is a commutative algebra, then $A^{+}$ will also be commutative.

Related concepts

• unit

• unitality