nLab unitalization

Idea

The unitalization of a non-unital algebra is a unital algebra with a “unit” (an identity element) freely adjoined.

This is a free functor: Rng $\to$ Ring.

Definitions

For (non)associative algebras

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

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

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

$$(a_1, 0) (a_2, 0) = (a_1 a_2, 0) ,$$

$$(0, r_1) (0, r_2) = (0, r_1 r_2) ,$$

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

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 $(a, r)$ 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.

For other (special) cases

See

• unitisation of rings;

• unitisation of C*-algebras.

Properties

The unitalization functor is not a conservative functor (Andruszkiewicz).

However, it does become conservative when restricted to the full subcategory of nonunital algebras over a field $k$ that do not admit a surjective homomorphism to $k$ (Andruszkiewicz).

For $\mathbb{E}_k$-algebra

Unitisation in the generality of Ek-algebra – hence for nonunital Ek-algebras – unitalization is the content of (Lurie, prop. 5.2.3.13).

Related concepts

• unit

• unitality

References

• Jacob Lurie, section 5.2.3 of Higher Algebra

• Ryszard R. Andruszkiewicz, Surprise at Adjoining an Identity to an Algebra, Vietnam Journal of Mathematics, doi.