proof of product of left and right ideal
Proof. We must show that the difference of any two elements of is in , and that is closed under multiplication by . But both of these operations are linear in ; that is, if they hold for elements of the form , then they hold for the general element of . So we restrict our analysis to elements .
Clearly if , then by definition.
If , then
and thus is a two-sided ideal. This proves the first statement.
If are two-sided ideals, then since ; similarly, . This proves the second statement.
|Title||proof of product of left and right ideal|
|Date of creation||2013-03-22 17:41:25|
|Last modified on||2013-03-22 17:41:25|
|Last modified by||rm50 (10146)|