On weak Ikeda–Nakayama rings

A ring [Formula: see text] is called a left Ikeda–Nakayama ring (briefly, left IN-ring) if [Formula: see text] for all left ideals [Formula: see text] and all left ideals [Formula: see text] of [Formula: see text]. A ring [Formula: see text] is a called a left [Formula: see text]-ring if [Formula: see text] for all finitely generated semisimple left ideals [Formula: see text] and all left ideals [Formula: see text] of [Formula: see text]. It is clear that a left [Formula: see text] ring must be left [Formula: see text]. Right [Formula: see text]-rings can be defined similarly. It is shown that a left [Formula: see text]-ring may not be right [Formula: see text] and a left [Formula: see text] ring may not be left [Formula: see text]. One of the aims of this paper is to investigate left [Formula: see text]-rings satisfying additional conditions. We show that this weak injectivity property is useful in obtaining semiperfect rings. Moreover, we give several new characterizations of [Formula: see text] rings and [Formula: see text] rings via [Formula: see text]-rings. Finally, left [Formula: see text], [Formula: see text]-rings were considered.