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Let $(R,\mathfrak m)$ be a local noetherian ring, and $M$ an arbitrary $R$-module. Suppose that $\mathrm{Tor}_1(M,R/\mathfrak m)=0$. Does it follow that $M$ is flat? The answer is positive when $M$ …

All the previous answers send us to complicated examples since these are $1$-dimensional domains. But the OP has looked for A domain $D$ all of whose localizations $D_P$ for $P∈\mathrm{Spec}(D)$ …

Set $A=k[T,\sqrt f]$. In fact, $A=k[T,U]/(U^2-f)$. Then $A/gA\simeq k[T,U]/(g,U^2-f)$. If we set $L=k[T]/(g)$, then $A/gA\simeq L[U]/(U^2-\bar f)$. If $g$ ramifies in $A$ then there is $h\in k[T]$ suc …

Even strongly Laskerian rings are not necessarily coherent. By a theorem of Radu strongly Laskerian coherent rings are Noetherian, and there are examples of strongly Laskerian rings which are not No …

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$. In Macaulay's words, the exam …

This is a follow up of the question Example of a finitely generated faithful torsion module over a commutative ring on MathSE. Let $M$ be a finitely generated module over a commutative ring $R$ with …

The result of Bruns needed for determining the canonical module of the Rees algebra with respect to an ideal generated by a regular sequence is Theorem 8.8 in Bruns and Vetter, Determinantal Rings. Th …

Let $R$ be an integral domain, $K$ its field of fractions, and $I,J$ fractional ideals. If $R$ is a Krull domain, then $(R:_KIJ)=(R:_KI)(R:_KJ)$, or $(IJ)^{-1}=I^{-1}J^{-1}$. But I can't see any reaso …

Let $R$ be an integral domain, and $K$ its field of fractions. It is well known that for a finitely generated fractional ideal $I$ of $R$, and $S$ a multiplicative set we have $$(R:_KI)_S=(R_S:_KI_S). …

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them there are idealizations and valuation domains. But the first non-noetherian ring we are thinking …

Page 118, line 16, Example: Poincaré series should be $P(A,t)=(1-t)^{-s}l(A_0)$.

One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals? Let $I$ be a graded ideal in a polynomial ring over a fiel …

The quoted result relies on the following elementary characterization of local regular rings: Let $R$ be a local ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ …

I've posted an answer here.

The answer is definitely yes. The argument is very simple: the ring extension $$\mathbb Q[X_1,\dots,X_n]\subset \mathbb C[X_1,\dots,X_n]$$ is faithfully flat.