Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}\_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the center $\mathbf{Z}(\mbox{End}\_\mathbb{F}(\mathbf{V}))=\mathbb{F}$ implies that every maximal commutative subring $R\subset\mbox{End}\_\mathbb{F}(\mathbf{V})$ contains $\mathbb{F}$. What can $R$ be? Motivated by finite dimensions I have the following (naive) conjectures.


1. All such subrings R are isomorphic.
2. Every choice of a basis $\mathbf{V}\simeq\mathbb{F}^I$ gives rise to a set of commuting $\mathbb{F}$-linear projectors $p\_i(v)=v\_i$, $i\in I$. The subring generated by $\{p\_i\}\_{i\in I}$ over $\mathbb{F}$ is a maximal commutative subring.
3. All maximal commutative subrings $R$ arise in this way.


*Question:* Are all/some/any of these conjectures true? What would be a readable reference, i.e., free of abstract nonsense? Thank you.


**Edit:** In view of the comments below, the question needs to be reformulated.


1. Are all maximal commutative subrings isomorphic?
2. What is an example of a maximal commutative subring $R$? Thank you.



Maximal commutative subrings of the endomorphism ring of a vector space
Even for a $2$-dimensional vector space (so we're looking at subrings of the ring $M\_2(\mathbb{F})$ of $2\times2$ matrices over $\mathbb{F}$) there are nonisomorphic maximal commutative subrings. 


Both
$$\left\{\begin{pmatrix}x&0\\0&y\end{pmatrix}: x,y\in\mathbb{F}\right\}$$
and
$$\left\{\begin{pmatrix}x&y\\0&x\end{pmatrix}: x,y\in\mathbb{F}\right\}$$
are maximal commutative subrings, but they're not isomorphic.


