The MacWilliams extending property and pseudo-Frobenius rings with essential socles

In this paper, we study [Formula: see text]-MacWilliams right [Formula: see text]-modules. It is shown that [Formula: see text] the MacWilliams extension property for [Formula: see text] implies that [Formula: see text] is an automorphism-invariant module (the converse holds if [Formula: see text] is finitely cogenerated), [Formula: see text] for [Formula: see text], the MacWilliams extension property implies that [Formula: see text] is quasi-injective which yields that if the matrix ring [Formula: see text] over a right finitely cogenerated ring [Formula: see text] is right 1-MacWilliams, then [Formula: see text] is also right 1-MacWilliams. We also show that [Formula: see text] a right automorphism-invariant ring containing no infinite orthogonal sets of idempotents with essential right socle is right pseudo-Frobenius and [Formula: see text] if [Formula: see text] is a right [Formula: see text]-MacWilliams ring and it has ACC on right annihilators, then [Formula: see text] is quasi-Frobenius which generalize answers the question of when is a MacWilliams ring quasi-Frobenius and when is a quasi-Frobenius ring a right artinian, right automorphism-invariant?