Almost self-duality and Harada rings

The existence of self-duality for left Harada rings was investigated by J. Kado and K. Oshiro [J. Algebra 211 (1999) 384–408]. Recently the author constructed examples of left Harada rings without self-duality [J. Algebra 241 (2001) 731–744]. In this paper, we investigate almost self-duality and rings of a certain class, which contains right co-Harada rings (equivalently left Harada rings). Here almost self-duality is a generalization of self-duality. The main purpose of the paper is to show that every ring of the class (particularly every right co-Harada ring) has an almost self-duality.