Question

Let I be a proper ideal of a commutative ring R with 1. Prove that I is a maximal 3. (10 ideal of R if and only if for every a e R\I, I+(a) : {i+ ar i e I,rE R} = R.

thanks

Answer 1

is a proper ideal of and let
ag I

Then
I(a)
is also an ideal of such that $I\subsetneq I+\left \langle a \right \rangle\subseteq R$ which means we must have
I(a) R
because is maximal

So if is a proper maximal ideal, then for every
ag I
we must have
I(a) R

On the other hand if for proper ideal , and for every
ag I
we have
I(a) R
, we will show that must be maximal

Suppose it is not maximal. Then there exists some ideal such that
うrう1

Let
EJ-
(such an element must exist as $I\subsetneq J$)

Then
I(r)
must equal

But we also have
ГСЛ (т)С
so that $R=I+\left \langle x \right \rangle\subseteq J$

And so we must have $J=R$ which contradicts our assumption that

うrう1

Thus, our assumption that is not maximal is incorrect

So that must be maximal

Therefore, for a proper ideal, is maximal if and only if for every
ag I
we have
I(a) R

$\blacksquare$

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