Isomorphisms of the endomorphism semigroups op modules. II

In the present work we prove that if $R_1$ is an antimatrix ring (i.e . $R_1$ is not isomorphic to any matrix ring $S_n$, $n>1$, over a ring $S$) and if all projective modules over $R_2$ are free, then isomorphism $\Phi$ of multiplicative endomorphism semigroups of free modules is induced by a s.l.i. If $R_1$ and $R_2$ are ordered rings, $_{R_1}A_1$ and $_{R_2}A_2$ are free modules, $r(A_1)>1$, $D_1$ and $D_2$ are the multiplicative semigroups of all positive endomorphisms of the partially or­dered modules $A_1$ and $A_2$, $\Phi$ is an isomorphism of $D_1$ upon $D_2$ then $\Phi$ is induced by an orderly-semilinear isomorphism of $A_1$ upon $A_2$.