Cyclic module

Let $𝑅$ be a ring. A (left) module $𝑉$ over $𝑅$ is cyclic iff it is finitely generated by a single generator $𝑣 \in 𝑉$ , i.e. #m/def/module

𝑉 = ⟨ 𝑣 ⟩

for some $𝑣 \in 𝑉$ . As an immediate consequence, the multiplication map

𝜆 : 𝑅 ↠ 𝑉 𝑟 \mapsto 𝑟 𝑣

is an $𝑅$ -epimorphism with kernel $𝑅 A n n 𝑉$ , so $𝑉$ by the First isomorphism theorem, $𝑉$ is isomorphic to a quotient module of $𝑅$ .

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