.Help me in

Q1 : give an example of not commutative ring and did not have identity.

The set of all polynomials over the integer numbers with even free coefficient, wrt the usual operations of sum and multiplication of polynomials.

Q2: Suppose is ring and are Ideals in such that :

Prove is:

As stated the problem is false big time: just take the zero ideal to get a contradiction, with any nonzero ring and any non-trivial proper ideal.

Check carefully the conditions...

Tonio