Are all ideals of a ring rings themselves? A question

I've two proofs that I'm having difficulty in proving.

**Problem:**

A nonempty subset of a ring is called an ideal of if is closed with respect to addition and negatives and absorbs products in .

Pinter's Abstract Algebra book on page 187-188:

Let be a ring, and an ideal of . For any element , the symbol denotes the set of all sums , as remains fixed and ranges over . That is,

is called a coset of in .

There is a way of adding and multiplying cosets which works as follows:

And

**My question:**

How do I prove that:

And

?

How did the author figure out that left side is equal to right side for both of these equations? Is it possible to prove both of these equations?

Re: Are all ideals of a ring rings themselves? A question

Let .

Because J is an ideal and ideals are subring. So by definition of subring:

And so:

And for the second proof:

And:

So

Q.E.D