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4 votes
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A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
Bruce's user avatar
  • 41
4 votes
1 answer
364 views

Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$

Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread. Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
Salvo Tringali's user avatar
3 votes
0 answers
154 views

Prime ideals in $R \subseteq \mathbb{C}[x,y]$, $\dim(R)=2$

Prime ideals in $\mathbb{C}[x,y]$ were listed here; they are: (i) $(0)$. (ii) $(f)$, where $f$ is an irreducible polynomial. (iii) $(x-\lambda,y-\mu)$, where $\lambda,\mu \in \mathbb{C}$. Now let $R \...
user237522's user avatar
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3 votes
0 answers
245 views

Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal

I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
asrxiiviii's user avatar
3 votes
0 answers
158 views

A characterization for a commutative ring with a special intersection property for prime ideals

Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...
A. C. Biller's user avatar
2 votes
0 answers
82 views

Let $R$ be a non-catenary, and $f: R \to S$ be a finite monomorphism. Can $S$ be catenary?

Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \...
user 1's user avatar
  • 1,355
2 votes
0 answers
80 views

Prime elements in integrally closed extension of domains

Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$. Is $PS$ always a prime ideal of $S$? For classical examples of ...
Daniel W.'s user avatar
  • 365
2 votes
0 answers
70 views

Properties preserved in addition of ideals

If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?
Iqra Khan's user avatar