Monday, 27 April 2020

Intersection of subring

How to prove the intersection of two subrings is a subring?


Let S1 and S2  be two subrings of a ring R. Then S1∩S2 is not empty since at least 0⋿ S1∩S2.
Now in order to prove that S1∩S2 is subring, it is sufficient to prove that
1. a⋿ S1∩S2,b⋿ S1∩S2
a-b⋿ S1∩S2
2. a⋿ S1∩S2,b⋿ S1∩S2
ab⋿ S1∩S2
We have 
a⋿ S1∩S2
a⋿ S1,a⋿ S2
b⋿ S1∩S2
b⋿ S1,b⋿ S2

Now S1 and S2  are both subrings.

such that 
a⋿ S1,b⋿ S1
a-b⋿ S1 and ab⋿ S1
and a⋿ S2,b⋿ S2
ab⋿ S2.
Now a-b ⋿ S1,a-b⋿ S2
a-b⋿ S1∩S2
ab⋿ S1,ab⋿ S2
ab⋿ S1∩S2.
Thus 
a⋿ S1∩S2,
b⋿ S1∩S2
a-b⋿ S1∩S2.and
ab⋿ S1∩S2.
so that
 S1∩S2 is a subring of R.





Intersection of subring

How to prove the intersection of two subrings is a subring? Let S1 and S2  be two subrings of a ring R. Then S1∩S2 is not empty since ...