Let $({\cal{A}},{\cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${\operatorname{Ext}}_{\cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules.

Let $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

All rings are commutative with identity, and all modules are unital. In this paper we introduce the concept of a quasi-copure submodule of a multiplication $R$-module $M$ and will give some results about it. We give some properties of the tensor product of finitely generated faithful multiplication modules.

We introduce a class of ideals generated by a set of 2-minors of an (m × n)-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module ${{\mathbb{R}}^{\left( \mathbb{N} \right)}}$. We also prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover, we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper.

For a monomial ideal I of a polynomial ring S, a polarization of I is a square-free monomial ideal J of a larger polynomial ring such that S/I is a quotient of /J by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends ours sends it to x1y2y3 Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

Let R = ⊕i ∈ ℤRi be a ℤ-graded ring and M = ⊕i ∈ ℤMi be a graded R-module. Providing some results on graded multiplication modules, some equivalent conditions for which a finitely generated graded R-module to be graded multiplication will be given. We define generalised graded multiplication module and determine some of its certain graded prime submodules. It will be shown that any graded submodule of a finitely generated generalised graded multiplication R-module M has a kind of primary decomposition. Using this, we give a characterisation of graded primary submodules of M. These lead to a kind of characterisation of finitely generated generalised graded multiplication modules.

We will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M′ such that M′ ⊕ M″ s.t.r.f of degree k, when M″ s.t.r.f of degree k. Furthermore, we will show that, under certain conditions, if the completion of a module M s.t.r.f of degree k, then there is a non-negative integer k′ such that M s.t.r.f. of degree k′. Moreover, we state a corrected version of Leung and Man's theorem (K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293) on Noetherian rings that satisfies the radical formula.

Let R be a commutative ring with identity. We will say that an R-module M has Nakayama property, if IM = M, where I is an ideal of R, implies that there exists a ∈ R such that aM = 0 and a − 1 ∈ I. Nakayama's Lemma is a well-known result, which states that every finitely generated R-module has Nakayama property. In this paper, we will study Nakayama property for modules. It is proved that R is a perfect ring if and only if every R-module has Nakayama property (Theorem 4.9).

Let B be a submodule of an R-module M. The intersection of all prime (resp. weakly prime) submodules of M containing B is denoted by rad(B) (resp. wrad(B)). A generalisation of 〈E(B)〉 denoted by UE(B) of M will be introduced. The inclusions 〈E(B)〉 ⊆ UE(B) ⊆ wrad(B) ⊆ rad(B) are motivations for studying the equalities UE(B) = wrad(B) and UE(B) = rad(B) in this paper. It is proved that if R is an arithmetical ring, then UE(B) = wrad(B). In Theorem 2.5, a generalisation of the main result of [11] is given.

An $R$-module $M$ is called a multiplication module if for each submodule $N$ of $M,\,N\,=\,IM$ for some ideal $I$ of $R$. As defined for a commutative ring $R$, an $R$-module $M$ is said to be semiprimitive if the intersection of maximal submodules of $M$ is zero. The maximal spectra of a semiprimitive multiplication module $M$ are studied. The isolated points of $\text{Max}\left( M \right)$ are characterized algebraically. The relationships among the maximal spectra of $M$, $\text{Soc}\left( M \right)$ and $\text{Ass}\left( M \right)$ are studied. It is shown that $\text{Soc}\left( M \right)$ is exactly the set of all elements of $M$ which belongs to every maximal submodule of $M$ except for a finite number. If $\text{Max}\left( M \right)$ is infinite, $\text{Max}\left( M \right)$ is a one-point compactification of a discrete space if and only if $M$ is Gelfand and for some maximal submodule $K$, $\text{Soc}\left( M \right)$ is the intersection of all prime submodules of $M$ contained in $K$. When $M$ is a semiprimitive Gelfand module, we prove that every intersection of essential submodules of $M$ is an essential submodule if and only if $\text{Max}\left( M \right)$ is an almost discrete space. The set of uniform submodules of $M$ and the set of minimal submodules of $M$ coincide. $\text{Ann}\left( \text{Soc}\left( M \right) \right)M$ is a summand submodule of $M$ if and only if $\text{Max}\left( M \right)$ is the union of two disjoint open subspaces $A$ and $N$, where $A$ is almost discrete and $N$ is dense in itself. In particular, $\text{Ann}\left( \text{Soc}\left( M \right) \right)\,=\,\text{Ann}\left( M \right)$ if and only if $\text{Max}\left( M \right)$ is almost discrete.

In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property $(\text{WLP})$, and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having $\text{WLP}$, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have $\text{WLP}$. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has $\text{WLP}$ but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double $G$-linkage, the implementations use very different methods.

Finitely generated modules with finite $F$-representation type over Noetherian (local) rings of prime characteristic $p$ are studied. If a ring $R$ has finite $F$-representation type or, more generally, if a faithful $R$-module has finite $F$-representation type, then tight closure commutes with localizations over $R$. $F$-contributors are also defined, and they are used as an effective way of characterizing tight closure. Then it is shown that $\lim_{e \to \infty} ({\#(\e M,M_i)}/ {(ap^d)^e})$ always exists under the assumption that $(R,\m)$ satisfies the Krull–Schmidt condition and $M$ has finite $F$-representation type by $\{ M_1, M_2, \dots, M_s \}$, in which all the $M_i$ are indecomposable $R$-modules that belong to distinct isomorphism classes and $a=[R/\m:(R/\m)^p]$.

An R-module is a simple divisible module if it is a nonzero divisible module that has no proper non-zero divisible submodules. We study simple divisible modules and their endomorphism rings, give some examples and determine all simple divisible modules over some classes of rings.