Example 1

Let V be an infinite-dimensional vector space over a commutative field K. Let End$^f_K(V)$ be the K-subalgebra of End$_K(V)$ consisting of the endomorphisms of finite rank. Let W be a linear subspace of V, and let $\mathfrak{a}_W$ be the set of elements of End$^f_K(V)$ whose kernel contains W; it is a left ideal of End$^f_K(V)$. An element $u$ of End$^f_K(V)$ is a right unit modulo $\mathfrak{a}_W$ if and only if we have $u(x) =x$ for every $x\in W$. Such an element $u$ exists, that is, $\mathfrak{a}_W$ is regular, if and only if W is finite-dimensional. The ideal $\mathfrak{a}_W$ is maximal and regular if and only if W has dimension 1.