ideal of an algebra
Let be an algebra over a ring .
Definition - A left ideal of is a subalgebra such that whenever and .
Equivalently, a left ideal of is a subset such that
, for all .
, for all and .
, for all and
Similarly one can define a right ideal by replacing condition 3 by: whenever and .
A two-sided ideal of is a left ideal which is also a right ideal. Usually the word ”” by itself means two-sided ideal. Of course, all these notions coincide when is commutative.
Since an algebra is also a ring, one might think of borrowing the definition of ideal from ring . The problem is that condition 2 would not be in general satisfied (unless the algebra is unital).
|Title||ideal of an algebra|
|Date of creation||2013-03-22 18:09:00|
|Last modified on||2013-03-22 18:09:00|
|Last modified by||asteroid (17536)|
|Synonym||left ideal of an algebra|
|Synonym||right ideal of an algebra|
|Synonym||two-sided ideal of an algebra|