question about semi-local ring and Artin ring.

Question

I don't understand this proof.

Before I read this,I think if R is not commutative,semilocal ring still has finitely many maximal left ideals.Because the following:

1.left Artin ring has finitely many maximal ideals.

2.There has a one to one correspndence between the maximal ideals of R which contains I and maximal ideals of R/I,where I is an ideal of R.

where am I wrong?I still don't understand the example given in the Remark.what is essential difference if we don't consider commutative ring.Thank you in advance!

Answer 1

You're wrong here:

left Artin ring has finitely many maximal [left] ideals.

I suppose you meant "maximal left ideals" since that is what the text is talking about.

For example, $M_2(\mathbb R)$ has infinitely many maximal left ideals. Consider the family of matrices of the form $A_\lambda = \begin{bmatrix}1&\lambda\\0&0\end{bmatrix}$ for $\lambda\in\mathbb R\setminus \{0\}$. The set of principal left ideals, each one generated by one of these matrices, forms a set of pairwise distinct maximal left ideals.

Obviously you can replace $\mathbb R$ with any infinite field and the argument still works.