Let $R$ be a commutative Noetherian local ring, $I$ an ideal of $R$, and let $M$ be a non-zero finitely generated $R$-module. In this paper, we establish some new properties of the local cohomology modules $H^i_I(M)$, $i\geq 0$. In particular, we show that if $(R,\mathfrak{m})$ is a Noetherian local integral domain of dimension $d \leq 4$ which is a homomorphic image of a Cohen-Macaulay ring and $x_1,\ldots,x_n$ is a part of a system of parameters for $R$, then for all $i\geq0$, the $R$-modules $H^i_{I}(R)$ are $I$-cofinite, where $I=(x_1,\ldots,x_n)$. Also, we prove that if $(R,\mathfrak{m})$ is a Noetherian local ring of dimension~$d$ and $x_1,\ldots,x_t$ is a part of a system of parameters for $R$, then $H^{d-t}_{\mathfrak{m}}(H^t_{(x_1,\ldots,x_t)}(R))\neq 0$. In particular, $\mu^{d-t}(\mathfrak{m},H^t_{(x_1,\ldots,x_t)}(R))\neq 0$ and ${\rm injdim}_R(H^t_{(x_1,\ldots,x_t)}(R))\geq d-t$.

Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence, we describe a family of non-monomial complete intersections that always have the strong Lefschetz property.

We explore resolutions of monomial ideals supported by simplicial trees. We argue that, since simplicial trees are acyclic, the criterion of Bayer, Peeva and Sturmfels for checking whether a simplicial complex supports a free resolution of a monomial ideal reduces to checking that certain induced subcomplexes are connected. We then use results of Peeva and Velasco to show that every simplicial tree appears as the Scarf complex of a monomial ideal and hence supports a minimal resolution. We also provide a way to construct smaller Scarf ideals than those constructed by Peeva and Velasco.

The notion of $\ppb \a$-boundedness is used to study several topics in the theory of separable abelian $p$-groups, including the torsion product of torsion-complete groups, the generalized core class property, and in the context of the constructible universe, groups whose socles are essentially finitely indecomposable as valuated vector spaces.

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I$ a homogeneous ideal in $S$ generated by a regular sequence $f_1,f_2, \ldots,f_k$ of homogeneous forms of degree~$d$. We study a generalization of a result of Conca et al.~\cite{CHTV} concerning Koszul property of the diagonal subalgebras associated to $I$. Each such subalgebra has the form $K[(I^e)_{ed+c}]$, where $c,e \in \mathbf{N}$. For $k=3$, we extend \cite[Corollary 6.10]{CHTV} by proving that $K[(I^e)_{ed+c}]$ is Koszul as soon as $c \geq {d}/{2}$ and $e >0$. We also extend \cite[Corollary 6.10]{CHTV} in another direction by replacing the polynomial ring with a Koszul ring.

Given a commutative ring $R$ with $1\neq0$, the zero-divisor graph $\Gamma(R)$ of $R$ is the graph whose vertices are the nonzero zero-divisors of $R$, such that distinct vertices are adjacent if and only if their product in $R$ is $0$. It is well known that the zero-divisor graph of any ring is isomorphic to that of its total quotient ring. This result fails for more general rings of quotients. In this paper, conditions are given for determining whether the zero-divisor graph of a ring of quotients of $R$ is isomorphic to that of $R$. Examples involving zero-divisor graphs of rationally $\aleph_0$-complete commutative rings are studied extensively. Moreover, several graph invariants are studied and applied in this investigation.

Many well-known closure operations such as integral closure and tight closure are both semiprime operations and Nakayama closures. In this short note, we begin the study on the overlap between Nakayama closures and semiprime operations. We exhibit examples of closure operations which are either semiprime or Nakayama but not the other. In the case of a discrete valuation ring we show that a closure operation $c$ is Nakayama if and only if it is semiprime and \[ (0)^c=\bigcap_{n \geq 1} (I^n)^c \] for any ideal $I$.