I'm reading the paper "the classification of algebras by dominant dimension" by Bruno J.Mueller, the link is here http://cms.math.ca/10.4153/CJM-1968-037-9.

In the proof of lemma 3 on page 402, there is a place I can't understand.

Who can tell me what $E_R \oplus * \cong \oplus X_R$ and $_AHom_R(E,X)\oplus * \cong \oplus _AHom_R(X,X)\cong \oplus_AA$ mean?