On weak co-$H$-sequences in two-sided Harada rings

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In [7] (and [8]) M. Harada studied a left artinian ring $R$ such that every non-small left $R$-module contains a non-zero injective submodule. And in [12] K. Oshiro called the ring a left Harada ring (abbreviated left $H$-ring) and further studied the ring in [13], [14], [15], $\ldots$. We can see many results on left Harada rings in [4] and many equivalent conditions in [3, Theorem B]. In [5] we give new concepts “co-$H$-sequence” and “$H$-epimorphism” to study two-sided Harada rings. In this paper, we intruduce another new concept “weak co-$H$-sequence”. And we further study two-sided Harada rings.