SUBRING
Definition 1 - 11
Let ( R , + , . ) be a ring , and let ? ? S ? R . Then S is called a
Subring of R if ( S , + , . ) is itself a ring .
Examples
{ 0 } is subring of Ze , { 0 } , Ze , and Z are subrings of Z .
( { [ 0 ]} , +4 , .4 ) , ( { [0] , [2] } , +4 , .4 ) , and ( Z4 , +4 , .4 ) are subrings
of ( Z4 , +4 , .4 ) .
Note
Every ring R has at lest two trivial subrings , {0} and R .
Theorem 1 – 12
A nonempty subset S of a ring (R, +, .) is a subring iff ? a , b ? R
we have a – b ? S and ab ? S .
Proof .
Suppose that S is subring , then ( S , + ) is commutative subgroup and
( S , . ) is semi-group , then if b ? S the additive inverse of b is -b ? S .
Then a –b = a + ( -b ) ? S .
Since ( S , . ) is semi-group , then S is closed . So ab ? S .
Conversely , since a + ( -b) = a –b ? S and S ? ? , then (S , + ) is an
additive subgroup of ( R , + ) . Since (R , + ) is commutative , then ( S , + )
is commutative .
Since S ? R , and . is associative on R , then . is associative on S ,
also ab ? S , then ( S , . ) is semigroup .
Because the two distributive laws hold in R , they also hold in S .
Hence , ( S , + , . ) is a subring . 8
Theorem 1 - 13
The intersection of two subrings is subring .
Proof .
It is clear .
Notes
The zero element of a subring is that of the ring
The additive inverse of an element of the subring is the same
as its inverse as a member of the ring .
In a ring with identity , a subring need not contain the identity
element .
Some subrings has a multiplicative identity , but the ring dose
not.
Both the ring and one of its subrings possess identity element ,
but they are distinct .The identity for the subring is necessarily
a divisor of zero in the larger ring