Isomorphism Classes of Algebras with Radical Cube Zero II

Abstract

We study to classify, up to isomorphism, algebras over a field k such that the radical cubed is zero and modulo the radical is a product of copies of k. The number of local quasi-Frobenius k-algebras with the condition is shown to be not less than the cardinality of k. In particular, the canonical forms of those algebras of dimension 5 are presented and their isomorphism classes are completely determined under some conditions on k.

A Construction of Local QF Rings and Its Characterization

“…In the former article, which is included in Chapter 10 of the book [2], several new examples of local graded QF rings with Jacobson radical cubed zero are given, and in the later book, rings with Jacobson radical cubed zero are discussed in a general theory of QF rings. In [5][6][7] we also studied the classification of local QF algebras over a field. In [10,11], QF rings and algebras are studied in connection with the associated graded ring.…”