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The following seems to be implied by most of the direct comments to OP's question, but I prefer to voice it, loud and clear: The answer is almost yes. Given $\varepsilon > 0$, we can find suc …

Here is a necessary and sufficient condition for $G^{-}(f)$ to be non-empty, taken from [1, Exercise 6.21]: Let $S = \begin{pmatrix} a & b/2 \\ b/2 & c \end{pmatrix}$ with $a,b$ and $c \in \mathbb{Z …

Yes, this is true and it follows immediately from [1, Corollary 4] Two numerical semigroups with multiplicity three are equal if and only if they have the same Froebenius number and the same …

The answer is yes. Claim. Let $R$ be an integral domain and let $f(X_1, \dots, X_n), g(X_1, \dots, X_n) \in R[X_1, \dots, X_n]$ $(n \ge 1)$ be polynomials with no non-constant common divisor. Then $f …

Here is a strengthening of the result established by Will Sawin: Claim 1. Let $R$ be a commutative ring with identity. Let $n$ and $k$ be positive integers and let $\prec$ be a monomial order on $R[X …

L. N. Vaserstein's theorem [2] asserts that if $R$ is a Dedekind ring of arithmetic type with infinitely many units then $SL_2(R) = E_2(R)$ holds. In other words $R$ is a $GE_2$-ring in the sense of …

Some authors add the requirement that a Dedekind domain not be a field. I will assume that a Dedekind domain is not a field, since otherwise any field is a counter-example. The answer is no: there are …

The rank of $\ker f$ is $n - k$ where $k$ is the free rank of $f(\mathbb{Z}^n)$. The following remark splits the computation of $k$ into two parts; the first can be effectively carried out provided th …

I'll begin with a general remark which will be illustrated by a computation in an arbitrary order of quadratic number field. If $\overline{I}$ contracts to $I$, i.e., if $\overline{I} \cap R = I$, the …

The goal of my answer is to elaborate on steps (1) and (2) of the accepted answer of Steven Landsburg. In particular, I would like to make clear how a "small enough ideal" looks like and how to produc …

This is a long comment posted as an answer on OP's request. The first three claims below are known to the OP. They serve as an introduction to our Claims 4 and 5. Our claims describe a wider set of m …

The answer is no: it is not possible, in general, to reduce a matrix over a principal ideal domain (PID) to a diagonal (or trigonal) matrix by means of elementary row and column operations. (This topi …