Four submodules suffice for realizing algebras over commutative rings

AbstractA positive answer is given to the infinite rank four submodules problem. Let λ be an infinite cardinal, and let A be an algebra over the commutative ring R which can be generated by no more than λ elements. Then there exist four R-submodules of the free A-module of rank λ such that the R-endomorphisms leaving the four submodules invariant are precisely the scalar multiplications by the elements of A