Lattice isomorphisms of modules over regular rings (Q1184586)

Let \((F,A)\) denote a unital left module over a ring \(F\), let \(E(F,A)\) be its endomorphism ring (operating on the right) and let \({\mathcal S}(A)\) be the partially ordered set of (direct) summands of \(A\). By a ``point'' of \((F,A)\) will be meant a summand isomorphic to \(F\). The author proves: 1) Let \((F,A)\) and \((G,B)\) be modules with \((F,A)\) free on at last three generators, and \(E(F,A)\) and \(E(G,B)\) regular rings. Let \(\alpha\) be an isomorphism of \({\mathcal S}(A)\) upon \({\mathcal S}(B)\). Then \(\alpha\) is induced by a semi-linear module isomorphism of \((F,A)\) upon \((G,B)\) if and only if \(\alpha\) maps some point of \(A\) onto a point of \(B\); 2). Let \((F,A)\) and \((G,B)\) be modules with \((F,A)\) free on at least two generators. Let \(\beta\) and \(\gamma\) be semi-linear isomorphisms of \((F,A)\) onto \((G,B)\). The following are equivalent: (i) \(\beta^{-1}\sigma\beta=\gamma^{- 1}\sigma\gamma\), for each \(\sigma\in E(F,A)\). (ii) \(S^ \beta=S^ \gamma\), for each summand \(S\subseteq A\). (iii) There exists a unit \(d\in G\) such that \(a^ \beta=da^ \gamma\) for each \(a\in A\).