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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 …

I gave a short answer to this question here: https://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators. A more general answer says the following: if $K$ is a field, $ …

I've posted an answer on M.SE.

I've posted an answer here.

I will try a quick answer based on the following well known Theorem. (Graded Noether Normalization) Let $K$ be a field and $S$ a graded $K$-algebra. Set $r=\dim S$. For homogeneous elements $p_1,\dot …

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 …

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 …

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). …

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 …

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 …

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 $R$ …

Set $R=k[p_1,\dots,p_t]$, $S=k[x_1,\dots,x_r]$ and $\mathfrak{m}=(x_1,\dots,x_r)$ (the maximal irrelevant ideal of $S$). We know that the ideal $(p_1,\dots,p_t)S$ is $\mathfrak{m}$-primary. A graded v …

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 …

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.