In this short note, we extend Faugére’s F4-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F4-algorithm is based on the simultaneous reduction of several polynomials.

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $\left(I^{k+1}{:}_{R}I\right)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $\left(I^{(k+1)}{:}_R{I}^{\left(1\right)}\right)=I^{\left(k\right)}$ for all $k$, where $I^{\left(k\right)}$ denotes the $k$th symbolic power of $I$. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial ideal has the...