# Tagged Questions

An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ...

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### Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R)$ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
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### Intersection of any set of ideals is an ideal

Prove that the intersection of any set of Ideals of a ring is an Ideal. I'm looking for hints. Let A, B both be Ideals of a ring R. Suppose $I \equiv A\cap B$. Since A and B are both Ideals of a ...
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### Proof Verification of Result Involving Maximal Ideals

In further investigation of a question I asked earlier, I came across the following result, the proof of which I hope can be looked over here. I personally find it kind of interesting and I hope ...
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### $R=\{ m+nr\sqrt{2} \mid m,n \in \Bbb Z \}$ and $I_{a,b}=\{ ma+n(b+r\sqrt{2}) \mid m,n \in \Bbb Z \}$

Let $r$ be a natural number and $R=\{ m+nr\sqrt{2} \mid m,n \in \Bbb Z \}$. We can show that $R$ is a subring of the ring $\Bbb Q [\sqrt{2}]$. My questions are as follows: $(1)$ Suppose that a ...
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### Can a Nonzero Element in $\mathbb Z[\omega]$ be Divisible by Arbitrarily Large Powers of $1-\omega$.

Question. Let $p$ be a prime and $\omega$ be a primitive $p$-th root of unity. Let $a$ be a nonzero element of $\mathbb Z[\omega]$. Can it happen that for each $n\in \mathbb N$, $(1-\omega)^n$ ...
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### Prime and maximal ideal

If I have to show that ideal A is not maximal, is it enough to show that A is not prime because it is usually easier? Every maximal ideal is prime so if we have ideal that is not prime, it can not be ...
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### Some ideal property in a local ring

If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question? Again, the new local ring ...
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### Ideal generated by given integers verification.

The question reads: Find the positive generator of the smallest ideal in $\mathbf Z$ containing the following ideals: a. $(4)$ and $(18)$. My answer is $(m)=(4)$. b. $(6)$ and $(35)$. My ...
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### If I and J are isomorphic ideals of a ring R, does it follow that $R/I \simeq R/J$?

The title pretty much sums it up. We know that $R/I \simeq R/J$ does not necessarily imply $I \simeq J$. But does the converse hold? I can't find any counterexample and all my efforts in proving it ...
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### On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR
Convention : All rings are commutative with unity unless stated otherwise. By ideals we will mean to include $\{0\}$ and $R$ also. Let us call an integer $n>1$ a "principal number" if any ring $R$...
### On those integers $n>1$ such that there exists a commutative ring with identity with exactly $n$ ideals
Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, \$\...