Rings-Algebra

What are Rings?

Let R is non-empty set with two binary operations called addition and multiplication. It is denoted by(+,.),  for all a,b⋿R we have a+b⋿Rand a.b⋿R  then this algebraic structure(R,+,.) is called a ring.

Ring with unity

If ring R exists by1 such that 1.a=a=a.1 all a⋿R, then R is called a ring with the unit element.

Commutative Ring

If a ring R, the multiplication composition is also commutative, if we have a.b=b.a all a,b⋿R, then Ris called commutative ring.

Properties of Rings

If R is a ring for all a,b,c⋿R

a0=0a=0

a(-b)=-(ab)=(-a)b

(-a)(-b)=ab

a(b-c)=ab-ac

(b-c)a=ba-ca

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.