unities of ring and subring
Let be a ring and a proper subring of it. Then there exists five cases in all concerning the possible unities of and .
and have a common unity.
has a unity but does not.
and both have their own non-zero unities but these are distinct.
has no unity but has a non-zero unity.
Neither nor have unity.
Note: In the cases 3 and 4, the unity of the subring must be a zero divisor of .
The ring and its subring have the common unity 1.
The subring of even integers of the ring has no unity.
Let be the subring of all pairs of the ring for which the operations “” and “” are defined componentwise (i.e. etc.). Then and have the unities and , respectively.
Let be the subring of all pairs of the ring (operations componentwise). Now has the unity but has no unity.
Neither the ring (operations componentwise) nor its subring consisting of the pairs have unity.
|Title||unities of ring and subring|
|Date of creation||2013-03-22 14:49:37|
|Last modified on||2013-03-22 14:49:37|
|Last modified by||pahio (2872)|