Because J is an ideal and ideals are subring. So by definition of subring:
And for the second proof:
I've two proofs that I'm having difficulty in proving.
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:
How do I prove that:
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?