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

\bf A^{+}=\bf R\oplus\bf A

The product operation is defined as follows:

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

The unity of $\bf A^{+}$ 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 one has a ring of finite characteristic $k$ , one can instead regard it as an algebra over $\mathbb{Z}_{k}$ and unitize accordingly to obtain a ring of characteristic $k$ .

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

Title	unitization
Canonical name	Unitization
Date of creation	2013-03-22 14:47:36
Last modified on	2013-03-22 14:47:36
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 16-00
Classification	msc 13-00
Classification	msc 20-00
Synonym	minimal unitization