Let M be a maximal ideal of a commutative ring R with identity. Prove that for each is a meximal ideal of R.
Proof so far.
Suppose that , and I want to show that J = R.
Or should I process another way, prove that J = ?
your proof is not correct, because in general: the reason is that an element of the LHS is in the form but an element of the RHS is in the form
so in general we only have: here's how to solve your problem: first if then we'll have:
because on the other hand we have: thus:
this proves half of your problem. now suppose then since we
have we must also have hence: for some so the constant coefficients in
both sides must be equal. thus: