In this short paper we consider the conjecture that for a finite dimensional commutative nilpotent algebra M over a perfect field of prime characteristic p, dimM > pdimMŵ here M p is the subalgebra of M generated by x p ,x € M. We prove that for any finite dimensional nilpotent algebra M (not necessarily commutative) over any field of prime characteristic p, dim M > p dim M^ for dim M^ < 2.

A spectral inclusion for block triangles is extended to "spectral" triangles.

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In this paper, we use properties of nilpotent rings to reprove an old theorem of R. Gilmer which classifies finite commutative primary rings having a cyclic group of units.