Abstract
Maximal commutative subalgebras of the algebra of n by n matrices over a field k very rarely have dimension smaller than n. There is a (B, N)-construction which yields subalgebras of this kind. The Courter's algebra that is of this kind was shown a (B, N)-construction where B is the Schur algebra of size 4 and N = k 4. That is, the Courter's algebra is isomorphic to B ⋉ (k 4)2, the idealization of (k 4)2. It was questioned how many isomorphism classes can be produced by varying the finitely generated faithful B-module N. In this paper, we will show that the set of all algebras B ⋉ N 2 fall into a single isomorphism class, where B is the Schur algebra of size 4 and N a finitely generated faithful B-module.
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