Almost self-duality and Harada rings.
Recall that a ring \(A\) is right Morita dual to a ring \(B\) if there exists a bimodule \(_BU_A\) that defines a Morita duality. In case \(A\) is right Morita dual to \(A\) itself, \(A\) has a self-duality. We say that \(A\) is right almost dual to \(B\) if there exist rings \(A_1=A\), \(A_2\), \(A_3,\dots,A_m\), \(A_{m+1}=B\) such that each \(A_i\) is right Morita dual to \(A_{i+1}\). In case \(A\) is right almost dual to \(A\) itself, \(A\) is said to have an almost self-duality. It is shown that such a ring \(A\) can be characterized by using the term of PF rings. Recall that a module \(X\) is non-small if \(X\) is not small in an injective hull of \(X\). Otherwise we call \(X\) small. A left Artinian ring with \[ \text{Every non-small left module contains a nonzero injective submodule}\tag{*} \] is called a left Harada ring. The existence of self-duality for left Harada rings was investigated by \textit{J. Kado} and \textit{K. Oshiro} [J. Algebra 211, No. 2, 384-408 (1999; Zbl 0926.16019)]. All the left Harada rings do not always have a self-duality. Indeed, the author of the paper under review constructed examples of left Harada rings without self-duality [ibid. 241, No. 2, 731-744 (2001; Zbl 1008.16020)]. However, he shows in the paper under review that every left Harada ring has an almost self-duality.
scientific article; zbMATH DE number 1867983
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