Sum of Finite Set of Submodules is Supremum of Lattice of Submodules

Theorem

Let $R$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $M = \struct {G, +, \circ}_R$ be an $R$-module.

Let $H_1, H_2, \ldots, H_n$ be submodules of $M$.

Then $H_1 + H_2 + \cdots + H_n$ is the supremum of $\set {H_1, H_2, \ldots, H_n}$ in the complete lattice of all submodules of $M$.

Proof

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Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.2$: Corollary