Commutative Algebras with Radical Cube Zero

Abstract

We present ''canonical forms'' of finite dimensional (quasi-Frobenius) commutative algebras L over a field k such that the radical cubed is zero and L modulo the radical is a product of copies of k. We also determine the isomorphism classes of the algebras L over some typical fields. Let X, Y 2 M n (k). Then X is said to be congruent to Y if there exists a P 2 GL n (k) such that X ¼ PY t P, where t P is the transpose of P.

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.…”

Isomorphism of Qf Algebras Over ℚ

“…In [1] we studied commutative local algebras over a field k with radical cubed zero and classified them under some conditions when the order of the multiplicative group k * of k modulo (k * ) 2 , the subgroup of square elements of k * , is 1 or 2. This condition for k is satisfied by a large class of fields, e.g.…”

“…To show(1), it suffices by Theorem 2.1 and a similar argument in the proof of Lemma 3.4 to prove that there exists e p ∈ Q * p satisfying Eqs. (3.3)-(3.5).…”