Recent development of Faith conjecture

Abstract

Is a semiprimary right self-injective ring a quasi-Frobenius ring? Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF: a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.

Keywords

Semiprimary right self-injective ring / Faith conjecture / division algebra D / (D / D)-space / Erdös-Kaplansky’s theorem / quasi-Frobenius ring / Nakayama permutation

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