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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

16857 questions
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$Im(f\otimes 1_M)=Im(f)\otimes M$ for flat module M

Suppose $N,N'$ are both $A$ modules and $f:N\rightarrow N'$ is an $A$ module homomorphism and $M$ is a flat $A$ module. How does one show that $Im(f\otimes 1_{M})=Im(f)\otimes M$, where $f\otimes 1_{M}:N\otimes M\rightarrow N'\otimes M$?
KnobbyWan
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A conjectural characterization of equal characteristic DVRs

Let $R$ be a commutative unital ring that is a DVR. Let $\mathfrak{m}$ be the unique maximal ideal of $R$. Assuming that $\mathrm{char}(R)=\mathrm{char}(R/\mathfrak{m})$, does there necessarily exist a homomorphism $R/\mathfrak{m}\rightarrow R$…
user251240
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A property of Cohen-Macaulay modules.

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay (CM) local ring of dimension $d>0$. Suppose that $M$ is a finitely generated CM $R$-module of $\dim_R M=i
Dat234
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Find the associated primes of the following ring

Find the associated primes of $\dfrac{\mathbb{C}[x,y,z]}{\langle xy,yz\rangle}$. I have already found that $ $ and $$ are associated primes. But i am not being able to find the others.
snehashis mukherjee
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Isomorphisms between rings of fractions of a ring

If $f: A\to B$ is a ring homomorphism between commutative rings and $S$ any multiplicative closed subset of $R$. Let $T=f(S)$. Then it is well known that there exists an $S^{-1}R$-module isomorphism $h:S^{-1}B\to T^{-1}B$, $b/s\mapsto b/f(s)$, where…
waqas mahmood
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Zero divisor in chain of radical ideals

Let $R$ be a commutative ring with identity and suppose that $J = \bigcup I_\alpha$ is a chain of radical ideals such that $a \in R$ is a zero divisor in every $R/I_\alpha$. Is $a$ necessarily a zero divisor in $R/J$? I don't expect this to be…
Badam Baplan
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Direct Limit of $A_{f_i}$'s for $\langle f_1, ..., f_n \rangle = 1$ in $A$.

Let $A$ be a ring, and take $f_1, ..., f_n$ in $A$. Geometrically, it seems $A_{f_i}$ cover $A$ when $\langle f_1, ..., f_n \rangle = 1$ (in that case, for each point $\mathfrak{p}$, there must be at least one $f_i$ not contained in $\mathfrak{p}$,…
Ronald J. Zallman
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Concerning Serre’s Intersection multiplicity

I am trying to understand a statement in a proof. The setup is $(R,m)$ is a $3$-dimensional regular local ring with infinite residue field, $\mathfrak{p}$ is a height-$2$ prime ideal and $x$ is not in $\mathfrak{p}$. Also, $\mathfrak{p}^{(m)}$…
user656543
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Nilradical is the zero ideal in an integral domain.

Let $R$ be a commutative ring with unity. Let $I$ be an ideal of $R.$ Then I know that $$\operatorname {rad} ({I}) = \bigcap\limits_{\substack {p \supseteq I \\ \text {p prime}}} p.$$ Now let $I = (0).$ Then by the similar reasoning $$\operatorname…
little o
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Idempotence of integral closure operation of ideals

Integral closure of an ideal $I$ of a ring $A$ is defined as follows: $$I^{int}:=\{x\in A~ |~\exists~ n\in \mathbb{N}~\text{and}~a_i\in I^i~\text{for}~1\leq i\leq n~ ~\text{such that}~ x^n+\sum_{i=1}^{n}a_ix^{n-i}=0~\}$$ Since it is a closure…
Sam
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Jacobson Radical in Localization.

Let $A$ a ring comutative with 1 and $I\subset A$ an ideal. Set $S=1+I$ a multiplicative set. Show that $S^{-1}I$ is contained Jacobson Radical of $S^{-1}A$. Hint?
Euler
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When is a map integral under these conditions?

Let $R \rightarrow S$ be a map of rings. This map is integral if and only if $R_{p} \rightarrow S_{p}$ is integral for every prime $p \subset R$. This implies that $R_p/ p R_p \rightarrow S_p / p S_p$ is an algebraic extension. Also, by Nakayama's…
Ronald J. Zallman
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For $M$ any finitely generated $R$-module determine $E_R(M)$.

Let $R$ be a principal ideal domain with quotient field $K$. For $p$ any prime element of $R$ and $P$ the prime ideal it generates, show that $$ E_R\left(R/P\right) \cong K/pR_p \cong K/R_p \cong R[p^{-1}]/R $$ I have already done this part. For…
Username Unknown
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Definition of going-up and going-down maps

The exercise 5.10 of Atiyah's Commutative Algebra gives the definition of going-up map: A ring homomorphism $f:A\rightarrow B$ is said to have the going-up (resp. the going-down property) if the conclusion of going-up theorem (resp. the going-down…
Mike
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Is there an isomorphism between $ \text{Spec}(R_{\mathfrak{p}}) $ and the prime ideals of $ R $ which are contained in $ \mathfrak{p}$?

Suppose $ R $ is a ring, and $ \mathfrak{p} \in \text{Spec}(R). $ I have been told that $ \text{Spec}(R_{\mathfrak{p}}) \cong \lbrace \mathfrak{q} \in \text{Spec}(R)\;| \mathfrak{q} \subset \mathfrak{p} \rbrace, $ and I am trying to figure out…
Confused Student
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