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.

Abstract. Separation of the spectra of the diagonal elements of a block triangle corresponds to comparison with its fundamental projection.is a Banach algebra with block structure: thus [3] A and B are Banach algebras with identities and M and N are bimodules over A and B and everything is explained by formal matrix multiplication. An upper triangle in G is an element of the formIt is well known [2], [3], that of the three spectraeach is contained in the union of the other two: this extends more generally to "spectral triangles" [4]. Disjointness between the spectra of a ∈ A and b ∈ B, or significant subsets of them, has consequences expressible [3] in terms of a comparison between the operatorsis holomorphic near the spectra of a ∈ A and b ∈ B, thenIn particular if and only if S has disjoint diagonal spectrathen P is a holomorphic function of S:(1.3) P = f (S) with f ∈ Holo σ(S) .

A bounded linear operator T is said to be "left-right consistent" if the spectra of all the products ST and T S coincide. In this note we relate the associated "consistency spectrum" to Fredholm theory, and to the "fat boundary". (2010): 46H05, 47A05. Mathematics Subject Classification

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