I am exploring the non-unital algebra $M_\infty(K)$, which consists of all matrices over a field $K$ whose rows and oclumns are indexed by $Z^+ \times Z^+$ which have only finitely many nonzero entries.
I am trying to distinguish between two idempotents, the first are idempotents of the form $e_{ii}$ where $e_{ij}$ is the matrix unit with 1 in the $(i,j)$ position and zeroes elsewhere. The second family of idempotents are of the form $$\sum_{i = 1}^n e_{i1} \quad \text{ or} \quad \sum_{j = 1}^m e_{1j},$$ that is, idempotents which are some sum of matrix units from the first column or row.
According to Lam's First Course, Exercise 12.2*, both of these idempotents are primitive since, when we consider the elements of $M_\infty(K)$ as endomorphisms of some vector space $V$, their image is one-dimensional.
These idempotents are clearly distinct in their function (for example, the former family additively generates a set of local units for $M_\infty(K)$ while the latter does not. However, I cannot (up until now) find a ring-theoretic property (such as primitivity etc) which distinguishes between them.
Am I mistaken in my reading of Exercise 12.2*?