Quasi-hereditary algebras related to local algebras

For ⋀ a finite dimensional local algebra with radical N where Nn = 0 ≠ N n-1 define (as a right A-module), then A(⋀) is quasi-hereditary and it has a unique heredity ideal J 1(A).Assume ⋀ satisfies the right socle condition (the socle series and the radical series of ⋀⋀ coincide). We show that then the algebra Ai/J1 (Ai-1 ) is isomorphic to A i-1 where Ai = A (⋀/Ni ). for 1 ≤ i ≤ n. Moreover we determine the canonical module for A(⋀) and we show that the Ringel dual of A(⋀) is isomorphic to the endomorphism ring of as a left module, provided that ⋀ also satisfies the left socle condition.