Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. They are defined as ideals such that $ab\in I$ implies $a\in I$ or $b\in I$. A maximal ideal ideal is an ideal which is maximal w.r.t. inclusion.

In the ring of integers maximal and prime ideals coincide. They are the sets that contain all the multiples of a given prime number, together with the zero ideal.

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Intersection of a multiplicative set and an Ideal

Good day, Let $M$ be a multiplicative set and $I$ any Ideal, both in a noetherian Ring $R$. I want to know whether there is a way to decide whether $M \cap I = \emptyset$ and, if not, to find an element $m \in M \cap I$. I know that $R \setminus M$…
kolja
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An Ideal is Maximal

In $\mathbb Z[\sqrt{-5}]$, define the ideals $=\mathbf p \langle 2, 1+\sqrt{-5}\rangle $, $\mathbf q=\langle3, 1+\sqrt{-5}\rangle$, and $\mathbf r=\langle3, 1-\sqrt{-5}\rangle$. Show that each ideals are maximal. I have solved for…
bgj123
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Ring theory (about prime ideal)

How can we find the number of prime ideals of $Z_{10^5}$ ? It has been asked that the number of prime ideal of $Z_{10^5}$ is $a) 2 $ $b) 5 $ $c) 10 $ $d) 10000$
Mrutyunjay Meher
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