- #1

mathmari

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MHB

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Let $R$ be a commutative ring with unit.

We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ( $a\in R,x\in M$ ) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if $a=t, x=(x_1, x_2, x_3)$.

From the structure theorem for finitely generated $R$-module there is a list $p_1, p_2, \dots , p_n$ irreducible of $R$ and a list $k_1, k_2, \dots , k_n$ positive integers such that $$M\cong R/\langle p_1^{k_1}\rangle \oplus \dots \oplus R/\langle p_n^{k_n}\rangle$$

How can we find these lists? (Wondering)