Confusion about a result in Pierce's book Associative Algebras.

Question

I am confused about two Lemmas in the book Associative Algebras by R.Pierce (pages 43-44).

In Lemma d, since each right ideal of a can be considered an algebra over $A$ then if $N$ is a minimal right ideal of $A$, by lemma a (vi), $N$ is also a minimal ideal of some $N_i$.

However by definition of minimility of both ideals we get that $N=N_i$.

Where is my mistake ? Why do we only have an isomorphism between the two ideals ?

Answer 1

since each right ideal of a can be considered an algebra over $$

An algebra over $A$? In what sense? Algebras are usually not defined over noncommutative rings, and that is what your context allows for.

However by definition of minimility of both ideals we get that $=_$

You could only get equality if one was contained in the other, but there is no such guarantee. You can at best say they're isomorphic.

For example, in $A=M_2(\mathbb R)$, let $N_1$ be the subset with zeros on the bottom row, and $N_2$ be the subset with zeros on the top row. Both are minimal right ideals, unequal as subsets, but isomorphic to each other (and to every other minimal right ideal.)