Let $k$ denote an algebraically closed field of characteristic zero and let $B$ be an affine normal $k$-domain. Let $\delta$ be a locally nilpotent derivation on $B$ and let $A$ be the kernel of $\delta$. It is known that $A$ need not be affine over $k$. In this paper we show that even if $A$ is not affine, $A$ is almost affine in the sense that there is an affine normal $k$-domain $C$ and two nonzero elements $x,y$ in $C$ such that $A=C[1/x]\cap C[1/y]$.

What are all rings $R$ for which $R^{\times }$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$-group? We answer this question for finite-dimensional commutative $k$-algebras, finite commutative rings, modular group algebras, and path algebras. Two interesting byproducts of this work are a characterization of Mersenne primes and a connection to Dedekind’s problem.

We consider two variants of Abelian groups where (all) proper characteristic subgroups are isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups where all proper fully invariant subgroups are isomorphic, which are studied in details by the authors in Commun. Algebra (2015). In addition, we also examine those Abelian groups having at least one proper characteristic subgroup isomorphic to the whole group. We prove in these directions, by the use of concrete terminology, that any basic subgroup of a $p$-primary separable weakly $IC$-group remains a weakly $IC$-group, as well as that a torsion-complete $p$-group is a weakly $IC$-group if and only if some of its basic subgroups are weakly $IC$-groups. These results extend those obtained by Grinshpon and Nikolskaya in Tomsk State Univ. J. Math. & Mech. (2011, 2012) and in Commun. Algebra (2011), respectively.

The direct sum decomposition of a module $M$ in a direct sum $M=M_1\oplus M_2$ does not produce, in general, a decomposition of the lattice $\mathcal{ℒ}(M)$ of all submodules of $M$ in a direct product of lattices $\mathcal{ℒ}(M)=\mathcal{ℒ}(M_1)\times \mathcal{ℒ}(M_2)$. When this happens we say $M=M_1\oplus M_2$ is a lattice decomposition of $M$. These particular decompositions have special properties: for instance, $M_1$, and $M_2$, are complemented distributive submodules.

The main aim of this paper is to characterize, in terms of $\text{Supp}(M)$, when the module $M$, over a commutative ring $A$, have a lattice decomposition. Thus, we show that there is a one-to-one correspondence between lattice decompositions of $M$ and partitions of $\text{Supp}(M)$ in two closed under specialization subsets satisfying some extra properties. These extra properties are always satisfied whenever $A$ is noetherian ring; in that case each closed under specialization partition always produces lattice decomposition. In particular, we obtain that a module $M$ such that $A/\text{Ann}(M)$ is noetherian has a nontrivial lattice decomposition if, and only if, there exists a partition of the set of all prime ideals, minimal in $\text{Supp}(M)$, in two sets $D_1$ and $D_2$ such that ideals in $D_1$ are comaximal with ideals in $D_2$.

We prove that the lattice decomposition is a local property and also show several applications of the lattice decomposition to the module structure, as well as its behavior in relation to some module constructions, change of ring and ring extensions, …. On the other hand, if $M$ has a lattice decomposition, then the simple modules which are subfactors of $M$ produce a decomposition of $\sigma [M]$, the category of all modules subgenerated by $M$, in a product of two subcategories.

We characterise ideals in two-dimensional regular local rings that arise as ideals of maximal minors of indecomposable integrally closed modules of rank two.

Let $R=\mathbb{𝕂}[x_1,\dots ,x_n]$ and $I\subset R$ be a homogeneous ideal. We first obtain certain sufficient conditions for the subadditivity condition of $R/I$. As a consequence, we prove that if $I$ is homogeneous complete intersection, then the subadditivity condition holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on the subadditivity condition, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity condition holds for $R/I(G)$. We then study the strand connectivity of edge ideals and obtain several classes of graphs whose edge ideals are strand connected. Finally, we compute upper bounds for multigraded Betti numbers of several classes of edge ideals.

We study the Cohen–Macaulay property of a particular class of radical extensions of an unramified regular local ring having mixed characteristic.

We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime characteristic which define either weakly $F$-nilpotent or $F$-nilpotent local rings are open with respect to the Zariski topology whenever $R$ is $F$-finite or essentially of finite type over an excellent local ring.

We explore the behavior of the sectional genera of certain primary ideals in Noetherian local rings. We provide characterizations of Cohen–Macaulay local rings in terms of the sectional genera, the Cohen–Macaulay type, and the second Hilbert coefficients for certain primary ideals. We also characterize Gorenstein rings and quasi-Buchsbaum rings in terms of the sectional genera for certain primary ideals.

Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{ℛ}$ be the inverse limit of the standard-graded polynomial rings $K[x_1,\dots ,x_n]$, and let ${\mathcal{ℛ}}^b$ be the subring of $\mathcal{ℛ}$ consisting of elements with bounded denominators. In previous joint work with Erman and Sam, we showed that $\mathcal{ℛ}$ and ${\mathcal{ℛ}}^b$ (and many similarly defined rings) are abstractly polynomial rings, and used this to give new proofs of Stillman’s conjecture. In this paper, we prove the complementary result that $\mathcal{ℛ}$ is a polynomial algebra over ${\mathcal{ℛ}}^b$.