8 problems

Let $R$ be a commutative regular ring, let $M$ be a commutative, cancellative, torsion-free monoid without nontrivial units, and let $i$ be a nonnegative integer. For each natural…

Atomicity conjecture. If $M_r$ is atomic, then $\mathbb{Q}[M_r]$ is atomic.

An exponentially cyclic monoid is a Puiseux monoid in the class studied by Gotti; for such a monoid $M$, consider its monoid algebra $\mathbb{Q}[M]$ over the field of rationals. Go…

Let $q$ belong to the set … and let $M_q$ be the exponentially cyclic Puiseux monoid associated to $q$. Gotti's atomicity-ascent conjecture. If $M_q$ is atomic, then its monoid alg…

Let $M$ be a finite monoid, and let $k$ be a field. Write $kM$ for the monoid algebra of $M$ over $k$, and assume that the Cartan matrix of the monoid algebra $\mathbb C M$ is non-…

Let $A$ be a domain of dimension $n$, let $R=A[X_1,\ldots,X_m]$, and let $I$ be the ideal of $R$ generated by $(X_1,\ldots,X_m)$. In this case…

Let $k$ be a commutative ring, let $M$ be an abelian monoid, and let $\mathfrak c=(c_1,c_2,\ldots)$ be a sequence of integers with $c_i\ge 2$. For each $c\ge 2$, the dilation…

Let $\Bbbk$ be an algebraically closed field of arbitrary characteristic, let $Q$ be a finitely generated commutative monoid, and let $\Bbbk[Q]$ be its monoid algebra. Mesoprimary…