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 be an algebra over a ring with unity . Then, as a module, the unitization of is the direct sum of and :
The product operation is defined as follows:
The unity of is .
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 , one can instead regard it as an algebra over and unitize accordingly to obtain a ring of characteristic .
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.
|Date of creation||2013-03-22 14:47:36|
|Last modified on||2013-03-22 14:47:36|
|Last modified by||rspuzio (6075)|