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The structure and classification up to isomorphism of a naturally arising class of local rings is determined. Although we are primarily interested in the case of a finite residue field K, our results apply in fact over any field K of characteristic 6= 2. The problem is shown to be equivalent to that of classifying two-dimensional subspaces of M2(K) up to congruence, and it is in these terms that the question is addressed.

Congruence of two-dimensional subspaces in M2(K) (characteristic ≠2)

Congruence of two-dimensional subspaces in M₂(K) (characteristic ≠2)