Finitely generated module over a module-finite $K$ -monoid

Let $𝑇$ be a Module-finite $K$ -monoid and $𝑀$ be a finitely generated $𝑇$ -module. Then $𝑀$ is a finitely generated $K$ -module. #m/thm/module

Prove

Let ${𝑡 𝑖} 𝑚 𝑖 = 1$ be an $K$ -spanning set for $𝑇$ and ${𝑚 𝑖} 𝑛 𝑖 = 1$ be a $𝑇$ -spanning set for $𝑀$ . Then

{𝑡 𝑖 𝑚 𝑗 : 𝑖 \in ℕ 𝑚, 𝑗 \in ℕ 𝑛}

is an $K$ -spanning set for $𝑀$ , since any $𝑣 \in 𝑀$ may be expressed as an $𝑇$ -linear combination of $𝑚 𝑗$ 's and the coëfficients may then be expressed as linear combinations of $𝑡 𝑖$ 's.

#state/tidy | #lang/en | #SemBr

Finitely generated module over a module-finite K-monoid

Finitely generated module over a module-finite $K$ -monoid