Imbedding the rings
In this section, the concept of imbedding the rings has been explained. The definition and important theorem related to imbedding the rings are presented as follows.
Definition 1. A ring (R,+, •) is imbedded in a ring (R ,+ , • ) if there exists some subring (S,+ , • ) of (R ,+ , • ) such that (R,+, •) ? (S,+ , • ).
Now in the following theorem, one can understand imbedding of any ring in another ring with identity.
Theorem 1. Any ring can be imbedded in a ring with identity.
Proof : Let (R,+ , •) be an arbitrary ring and the Cartesian product .
R× Z = { (r , n) = r ? R , n ? Z } ,
where Z is the set of integers.
If addition and multiplication are defined in R × by
(a , n ) + (b , m ) = (a + b , n + m) ,
(a , n ) • ( b , m ) = ( a • b + ma + nb , nm ) ,
then the system (R × Z , + , • ) forms a ring . This ring has a multiplicative identity , (0 , 1) , for
(a , n) • (0 ,1) = (a • 0 + n0 , n ) = (a , n) ,
and similarly ,
(0 ,1) • (a , n) = (a ,n) .
Now , consider the subset R × 0 of R × Z. The elements of R × 0 are all the pairs of the form (a ,0) .
Since
(a ,0) – (b ,0) = (a- b , 0) and (a ,0) • (b ,0) = (a•b , 0) ,
it follows that the triple (R × 0 , + , •) is a subring of (R × Z , + , •) .
Now , it require to show that the (R , + , . ) ? (R × 0 , + , .) .
To prove that , define the function f : R ? R × 0 by
f(a)= (a ,0) .
Here, f is a 1-1 mapping of R onto the set R× 0 . And f preserves the addition and multiplication operations as follows :
f(a+b)= (a+b , 0) = (a ,0) +(b ,0) = f(a) + f(b),
f(a•b)= (a•b , 0) = (a ,0) • (b ,0) = f(a) • f(b) .
Therefore (R , + , •) ? (R × 0 , + , •) .
Thus , (R , + , •) is imbedded in (R ×Z , + , • ), a ring with identity.