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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.

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Wikipedia's definition of finitely generated algebra.

This is Wikipedia's definition of a finitely generated algebra: A finitely generated algebra (also called an algebra of finite type) is an associative algebra $A$ over a field $K$ where there exists a finite set of elements $a_1,…,a_n$ of $A$…
Huma Abid
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Field of fractions of PID R with infinitely many max ideals not a f.g. R-algebra

Let $R$ be a principal ideal domain, with field of fractions $F$. Show that if $R$ has infinitely many maximal ideals then $F$ is not finitely generated as an $R$-algebra.
Bernoulli
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Quotient of units in the formal power series ring

Let $k[[x,y]]$ be the ring of formal power series in two variables over a field $k$. A unit in $k[[x,y]]$ is of the form $a_0+f$ where $f\in k[[x,y]]$ and $a_0$ is a unit in $k$. I heard that the quotient of two units in $k[[x,y]]$ is again a unit.…
user349424
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Mori domain if and only if every $v$-ideal is of finite type?

In commutative alebra, I proved that in a Mori domain, every $v$-ideal(divisorial ideal) is of finite type. But converse is hard to me..I don't know it's correct. Someone help me plz
Silement
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A local ring with krull dimension $0$ consists of only units and nilpotent elements.

Let be $R$ a local conmutative ring of dimension 0, I want to prove that ring has only units and nilpotent elements. I managed to get that in the case of non-zero divisors are units. Nevertheless, I couldn't get that there are also nilpotent…
Hakim Ziyech
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Showing a morphism to an extension of a ring is a derivation.

Let $R$ be a $k$-algebra and let $(R',\phi)$ be an extension of $R$ by a the $k$-algebra $R'$ by the ideal $I \subseteq R'$ such that $I^2=(0)$. Let $B$ be another $k$-algebra and assume there exist two $k$-algebra morphisms $f_1, f_2: B \to R'$…
user7090
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projectivity of a module under a faithfully flat ring homomorphism of commutative rings

Let $R\to S$ be a faithfully flat ring map between commutative rings and $M$ any projective $S$-module. Is it true that $M$ is projective $R$-module? Here, is my attempt: Let $M\oplus X\cong S^{I}$, for some $S$-module $X$ and indexing set $I$.…
waqas mahmood
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Proving a map $M/\mathfrak{m}M\to M\otimes_S k$ is isomorphism.

Let $S$ be a local ring with maximal ideal $\mathfrak{m}$. I want to prove that if $M$ is a $S-$module, then $$M\otimes_S k\cong M/\mathfrak{m}M.$$ where $k=S/\mathfrak{m}$. To prove it I defined $\psi:M\to M\otimes_Sk$ by $\psi(m)= m\otimes 1$. It…
LeviathanTheEsper
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Algebras smaller than their underlying ring.

Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring $R$ to the…
Huma Abid
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Proving that the module of Kähler differentials is generated by the image of the derivation directly from the universal property

Let $B$ be an $A$-algebra and let $d:B\rightarrow \Omega_{B/A}$ be its $B$ module of Kähler differentials relative to $B$. By an explicit construction of $\Omega_{B/A}$, I know that the $A$-module generated by the image of $d$ is the entire…
Evangelion045
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Is this really a free resolution?

I was reading the book (Combinatorial Commutative Algebra) of Ezra Miller, and I got stuck when they give the example of the Koszul complex (which, in the book, is given by taking the reduced chain complex of $\Delta^n$, labeling the rows and…
LeviathanTheEsper
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Atiyah's proof of Weak Nullstellensatz (Corollary 5.24)

This has been asked before at least two different times, but I am still confused so I thought of asking it myself. The corollary goes like this Let $k$ be a field, $B$ afinitely generated $k$-algebra. If $B$ is a field, then it is a finite…
fhn
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Characterization of DVR's

Let $K$ be a field, and let $S$ be the set of local subrings of $K$. Put an order $\leq$ on $S$ where $A \leq B$ when $A \subset B$ and the maximal ideal of $A$ is sent into the maximal ideal of $B$. Then the valuation subrings of $K$ are the…
Ronald J. Zallman
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Describing the kernel of this surjection

Let $k$ be a field. For a finite dimensional $k$-vector space $V$, write $k[V]$ for the symmetric algebra, which is noncanonically isomorphic to some polynomial ring in finitely many variables. Take a short exact sequence of $k$-vector spaces $0…
Ronald J. Zallman
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Completion and $S^{-1} A$

Let $\mathfrak{p}$ be a prime ideal of a ring $A$. The completion $\hat{A}$ of $A$ with respect to its adic-topology is used to simplify $A$ beyond the localization $A_{\mathfrak{p}}$. For a multiplicative subset $S \subset A$, we have the…
Ronald J. Zallman
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