how to identify the ideals of a ring by using canonical homomorphism?

Question

Assume we have a quotient ring $R'=\mathbb{C}[t]/(t-1) $. How can I find the ideals of $ R' $ by using the cannonical homomorphism $ H$ from $\mathbb{C}[t] $ to $ R' $.

This is my homework actually but since I want to deal with it my self I quite modified the question.

score 1 · Answer 1 · answered Mar 12 '15 at 12:16

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This appears to be an application of the Correspondence Theorem. Find the ideals $I$ of $\mathbb{C}[t]$ such that $(t-1) \subseteq I \subseteq \mathbb{C}[t]$. (Hint: there aren't many.) The ideals of $R'$ will be of the form $I/(t-1)$,

answered Mar 12 '15 at 12:16

Andreas Caranti

68,873

1

How can I use the fact that the ring is PID? – Ulgen Mar 12 '15 at 12:21
Any ideal $I$ will be of the form $(a)$ for some $a$. Now use the fact that $(b) \subseteq (a)$ iff $a \mid b$. – Andreas Caranti Mar 12 '15 at 12:30

how to identify the ideals of a ring by using canonical homomorphism?

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