unitization

The operation of unitization allows one to add a unity element to an algebra. Because of this construction, one can regard any algebra as a subalgebra of an algebra with unity. If the algebra already has a unity, the operation creates a larger algebra in which the old unity is no longer the unity.

Let $𝐀$ be an algebra over a ring $𝐑$ with unity $1$. Then, as a module, the unitization of $𝐀$ is the direct sum of $𝐑$ and $𝐀$:

$${𝐀}^{+}=𝐑\oplus 𝐀$$

The product operation is defined as follows:

$$(x,a)\cdot (y,b)=(xy,ab+xb+ya)$$

The unity of ${𝐀}^{+}$ is $(1,0)$.

It is also possible to unitize any ring using this construction if one regards the ring as an algebra over the ring of integers (http://planetmath.org/Integer). (See the entry every ring is an integer algebra for details.) It is worth noting,however, that the result of unitizing a ring this way will always be a ring whose unity has zero characteristic. If onehas a ring of finite characteristic $k$, one can instead regard it as an algebra over ${\mathbb{Z}}_k$ and unitizeaccordingly to obtain a ring of characteristic $k$.

The construction described above is often called “minimal unitization”. It is in fact minimal, in the sense thatevery other unitization contains this unitization as a subalgebra.