symmetric monoidal (∞,1)-category of spectra
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 $R$ 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
or in general
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.
See
Unitisation in the generality of Ek-algebra – hence for nonunital Ek-algebras – unitalization is the content of (Lurie, prop. 5.2.3.13).