Zorn Lemma, opposite ring and so on........

Question

I just wanted to confirm some stuff with you regarding ideals, rings and the Zorn Lemma:

Given that

1) A right ideal of any ring automatically is a left ideal of its opposite ring

and

2) that every nonzero ring has a maximal right ideal (by Kuratowski-Zorn’s lemma),

Is it fair to conclude that the maximal right ideal of a nonzero ring is the maximal left ideal of its opposite?

If so, could you illustrate it with the example of a quadratic 2x2 matrix with zero entries in the bottom row?

Answer 1

I suggest you do the following:

• First, show that if $I$ is a left ideal in a ring $R$, then $I$ (the exact same set) is a right ideal in the opposite ring $R^{\mathrm{op}}$.

• Next, show that if $I$ is a maximal left ideal in a ring $R$, then $I$ is a maximal right ideal in the opposite ring $R^{\mathrm{op}}$.

It follows from this that the set of maximal left ideals in a ring $R$ is exactly the same set as the set of maximal right ideals in the opposite ring $R^{\mathrm{op}}$.