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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 $P∈\mathrm{Spec}(D)$ …

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 …

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.

This is a classical result. See, for instance the book of Iyengar, Leuschke, Leikin, Miller, Miller, Singh, Walther, Twenty-Four Hours of Local Cohomology, Remark 9.14.

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 …

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

Although the OP is mainly interested in noncommutative results and examples, let me say a few words about the commutative case. Let $k\subset K$ be a field extension. N. Bourbaki in Algebre. Chapitr …

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). …

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 …

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 …

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 …

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 …

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 a proof here for the special case when $M$ is cyclic. Furthermore, I've mentioned that the result holds for finitely generated modules when the sequence is $R$-regular and $M$-regular.

I've posted an answer here.