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Indeed, it is enough to check that each $S(g_i,g_j)$ can be reduced to zero in some way. Indeed, if you trace carefully the proof of Buchberger's criterion, you will see that one only needs the existe …

Trying to write down something along the lines you were looking for, I would argue that over an algebraically closed field the set of common zeros of $k < n$ polynomials has codimension at most $k$ an …

In the homogeneous case that is true, I believe. To show that $A=k[x_1,\ldots,x_n]$ is a finitely generated $k[f_1,\ldots,f_n]$-module, let's notice that for a regular sequence $f_1,\ldots,f_n$, the c …

If you consider $S$ as you propose, that is $\bigoplus\limits_{n_i,n_j=0}T_{\underline{n}}$, then $T$ is a bi-graded ring over $S$: $$T_{p,q}=\bigoplus\limits_{n_i=p,n_j=q}T_{\underline{n}}.$$ By th …

Assume $R$ is a PID. Clearly, there is a short exact sequence $0\to M_2\to A\to M_1\to 0$, where $M_1\subset R^r$ is the image of the map $A\to R^r$, so a free module of rank $d\le r$, and $M_2\cong …

Another formula (almost without alternating signs) can be obtained as a variation of the comment of David Speyer. Namely, for each $S\subset\{1,\ldots,k\}$ we can consider the set of all monomials tha …

It just crossed my mind that there is another way to compute the cardinality of the complement (and I decided to post it as well to demonstrate the power of generating functions): it is the coefficien …

A proof using Gröbner bases is in Using algebraic geometry by David A. Cox, John B. Little, Donal O'Shea, Theorem 2.1. However, I was always sure that there should be (at least in the graded case) an …

Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inve …

You might be interested in "Gröbner bases of ideals invariant under endomorphisms", by Vesselin Drensky and Roberto La Scala, Journal of Symbolic Computation (2006), Issue 7, Pages 835-846. In your sp …

This algebra $R$ is a quadratic Koszul algebra (in fact, it is easily seen to be PBW, that is has a quadratic Groebner basis), from which one can immediately construct a bimodule resolution; the bimod …

I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote hi …

What exactly you Googled? There are many standard references, e.g. "Gröbner bases and primary decomposition of modules", by Elizabeth W.Rutman (https://www.sciencedirect.com/science/article/pii/074771 …

Here (see the very last paragraph) it is stated that every matrix with trace zero over a PID is a commutator. However, I can't come up with a proof right away; the only proof for matrices over a field …

Suppose $x\in N$, so that $x^n=0$ for some $n$. Then using the product rule for derivations many times, we see that $$ 0=D^n(x^n)=n! D(x)^n+Y, $$ where $Y$ is divisible by $x$. Therefore, $D(x)^{n …