nLab unitalization




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.


For (non)associative algebras

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

The inclusion functor U:RAlg uRAlg nuU\colon R Alg_{\mathrm{u}} \to R Alg_{\mathrm{nu}} has a left adjoint () +:RAlg uRAlgu(-)^+\colon R Alg_{\mathrm{u}} \to R Alg{\mathrm{u}}. For ARAlg nuA \in R Alg_{\mathrm{nu}} we say A +A^+ is the unitalization of AA.

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

(a 1,0)(a 2,0)=(a 1a 2,0), (a_1, 0) (a_2, 0) = (a_1 a_2, 0) ,
(0,r 1)(0,r 2)=(0,r 1r 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 1a 2+r 2a 1+r 1a 2,r 1r 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)(a, r) as a+ra + r or ara \oplus r, which makes the above formulas obvious.

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

For other (special) cases



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 kk that do not admit a surjective homomorphism to kk (Andruszkiewicz).

For 𝔼 k\mathbb{E}_k-algebra

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


Last revised on April 24, 2021 at 19:27:32. See the history of this page for a list of all contributions to it.