We will prove that if $G$ and $H$ are modules over a principal ideal domain $R$ such that the endomorphism rings $\End_R(R\oplus G)$ and $\End_R(R\oplus H)$ are isomorphic, then $G\cong H$. Conversely, if $R$ is a Dedekind domain such that two $R$-modules $G$ and $H$ are isomorphic whenever the rings $\End_R(R\oplus G)$ and $\End_R(R\oplus H)$ are isomorphic, then $R$ is a PID.

We study the combinatorics of squarefree revlex ideals in a polynomial ring in $n$ variables with coefficients in a field $K$. We focus our attention on the relationship between squarefree revlex ideals and revlex ideals. Also, we analyze the extremal Betti numbers of squarefree revlex ideals.

We determine the defining equations of the Rees algebra of an ideal $I$ in the case where $I$ is a square-free monomial ideal such that each connected component of the line graph of the hypergraph corresponding to $I$ has at most $5$ vertices. Moreover, we show in this case that the non-linear equations arise from even closed walks of the line graph, and we also give a description of the defining ideal of the toric ring when $I$ is generated by square-free monomials of the same degree. Furthermore, we provide a new class of ideals of linear type. We show that when $I$ is a square-free monomial ideal with any number of generators and the line graph of the hypergraph corresponding to $I$ is the graph of a disjoint union of trees and graphs with a unique odd cycle, then $I$ is an ideal of linear type.

We compute the reverse lexicographic generic initial ideals of the powers of a $2$-complete intersection ideal $I$. In particular, we present six algorithms to compute these generic initial ideals; the power and the relative degrees of the minimal generators of $I$ determine the algorithm to be used.

It is proved that the Stanley conjecture holds for monomial ideals of mixed products, i.e., if $I$ is an ideal of mixed products in a polynomial ring $S$ over a field, then ${\rm sdepth}_S(I) \geq {\rm depth}_S(I)$ and ${\rm sdepth}_S(S/I) \geq {\rm depth}_S(S/I)$.

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the hafnian of a generic symmetric matrix are each generated in degree~2. As a consequence, using a result of Ranestad and Schreyer, we give lower bounds to the cactus rank and rank of each of these invariants. We compare these bounds with those obtained by Landsberg and Teitler.

In a recent work, Fouli and Lin generalized Villarreal's result and showed that if each connected component of the line graph of a squarefree monomial ideal contains at most a unique odd cycle, then this ideal is of linear type. In this short note, we reprove this result with Villarreal's original ideas together with a method of Conca and De Negri. We also propose a class of squarefree monomial ideals of linear type.