# Contents

## 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 $\left(-{\right)}^{+}:R{\mathrm{Alg}}_{\mathrm{u}}\to R\mathrm{Alg}\mathrm{u}$. 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

$\left({a}_{1},0\right)\left({a}_{2},0\right)=\left({a}_{1}{a}_{2},0\right),$(a_1, 0) (a_2, 0) = (a_1 a_2, 0) ,
$\left(0,{r}_{1}\right)\left(0,{r}_{2}\right)=\left(0,{r}_{1}{r}_{2}\right),$(0, r_1) (0, r_2) = (0, r_1 r_2) ,
$\left(a,0\right)\left(0,r\right)=\left(0,r\right)\left(a,0\right)=\left(ra,0\right),$(a,0) (0,r) = (0,r) (a,0) = (r a, 0) ,

or in general

$\left({a}_{1},{r}_{1}\right)\left({a}_{2},{r}_{2}\right)=\left({a}_{1}{a}_{2}+{r}_{2}{a}_{1}+{r}_{1}{a}_{2},{r}_{1}{r}_{2}\right).$(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 $\left(a,r\right)$ 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.

Revised on January 14, 2012 07:08:55 by MTS? (184.187.181.145)