All Questions
Tagged with ac.commutative-algebra ideals
8 questions
21
votes
4
answers
5k
views
The number of ideals in a ring
Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
- 1,881
5
votes
1
answer
1k
views
Ideal generated by two univariate, coprime, integer polynomials
Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
- 7,746
5
votes
1
answer
759
views
On the annihilator of a module
Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such
that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$?
Remark. The annihilator of a ...
- 303
3
votes
1
answer
257
views
Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?
This question might be below the level of MO, so apologies in advance. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring and $L(R)$ ...
- 2,605
2
votes
0
answers
77
views
On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down
Let $R$ be an integral domain with fraction field $K$ such that for every ring $R \subseteq S \subseteq K$ , the ring extension $R \subseteq S$ satisfies Going Down property i.e. for any chain of ...
user111524
1
vote
0
answers
140
views
On finite dimensional commutative algebras and regular sequences
Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
- 1,227
1
vote
1
answer
787
views
On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind
Let $R$ be an integral domain. Consider the set $$S := \big\{a \in R\smallsetminus \{0\}
: Ra+Rx \text{ is a principal ideal } \forall x \in R \big \}.$$ Is $S$ a saturated multiplicative closed ...
user111524
-2
votes
2
answers
763
views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492