Suppose $M$ and $N$ are finitely generated modules over some (possibly non-commutative) ring $R$. Let there be a lattice isomorphism between the lattice of submodules of $M$ and those of $N$ (i.e. their respective sets of submodules can be put into bijection with one another). Is this sufficient for there to be a $\textit{module}$ homomorphism between $M$ and $N$? If so, what conditions do $M$ and $N$ need to fulfill in order for this to be an injection, or a surjection?
2026-03-25 19:01:47.1774465307
Lattice Homomorphisms and Modules
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