Unstable modules with only the top k Steenrod operations

Unstable modules over the Steenrod algebra with only top $k$ operations are introduced in language of ringoids. We prove category such has homological dimension at most $k$. A pratical method, which generalizes $\Lambda$ complex, to compute $\mathrm{Ext}$ group from spheres is proposed. also able establish several functors relate and unstable algebra, describe connections between groups them.