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 sumPlanetmathPlanetmath of 𝐑 and 𝐀:


The product operation is defined as follows:


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 integersMathworldPlanetmath (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 β„€k and unitize accordingly to obtain a ring of characteristic k.

The construction described above is often called β€œminimal 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