For an ideal I in a Noetherian ring R, the Fitting ideals $\mathrm{Fitt}_j(I)$ are studied. We discuss the question of when $\mathrm{Fitt}_j(I)=I$ or $\sqrt{\mathrm{Fitt}_j(I)}=\sqrt{I}$ for some j. A classical case is the Hilbert–Burch theorem when $j=1$ and I is a perfect ideal of grade 2 in a local ring.

Let $(R,\mathfrak {m})$ be a Noetherian local ring and I an ideal of R. We study how local cohomology modules with support in $\mathfrak {m}$ change for small perturbations J of I, that is, for ideals J such that $I\equiv J\bmod \mathfrak {m}^N$ for large N, under the hypothesis that $R/I$ and $R/J$ share the same Hilbert function. As one of our main results, we show that if $R/I$ is generalized Cohen–Macaulay, then the local cohomology modules of $R/J$ are isomorphic to the corresponding local cohomology modules of $R/I$, except possibly the top one. In particular, this answers a question raised by Quy and V. D. Trung. Our approach also allows us to prove that if $R/I$ is Buchsbaum, then so is $R/J$. Finally, under some additional assumptions, we show that if $R/I$ satisfies Serre’s property $(S_n)$, then so does $R/J$.

A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

Let V be a finite dimensional vector space over the field with p elements, where p is a prime number. Given arbitrary $\alpha ,\beta \in \mathrm {GL}(V)$, we consider the semidirect products $V\rtimes \langle \alpha \rangle $ and $V\rtimes \langle \beta \rangle $, and show that if $V\rtimes \langle \alpha \rangle $ and $V\rtimes \langle \beta \rangle $ are isomorphic, then $\alpha $ must be similar to a power of $\beta $ that generates the same subgroup as $\beta $; that is, if H and K are cyclic subgroups of $\mathrm {GL}(V)$ such that $V\rtimes H\cong V\rtimes K$, then H and K must be conjugate subgroups of $\mathrm {GL}(V)$. If we remove the cyclic condition, there exist examples of nonisomorphic, let alone nonconjugate, subgroups H and K of $\mathrm {GL}(V)$ such that $V\rtimes H\cong V\rtimes K$. Even if we require that noncyclic subgroups H and K of $\mathrm {GL}(V)$ be abelian, we may still have $V\rtimes H\cong V\rtimes K$ with H and K nonconjugate in $\mathrm {GL}(V)$, but in this case, H and K must at least be isomorphic. If we replace V by a free module U over ${\mathbb {Z}}/p^m{\mathbb {Z}}$ of finite rank, with $m>1$, it may happen that $U\rtimes H\cong U\rtimes K$ for nonconjugate cyclic subgroups of $\mathrm {GL}(U)$. If we completely abandon our requirements on V, a sufficient criterion is given for a finite group G to admit nonconjugate cyclic subgroups H and K of $\mathrm {Aut}(G)$ such that $G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.

The existence of Ulrich modules for (complete) local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether complete local domains always have Ulrich modules. In this paper, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences – a weaker notion recently introduced by Ma – in the negative. We construct many local domains in all dimensions $d \geq 2$ that do not have any Ulrich modules. Moreover, we show that when $d = 2$, these local domains do not have weakly lim Ulrich sequences.

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$. This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$. Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$. Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

In this note we prove the following surprising characterization: if $X\,\subset \,{{\mathbb{A}}^{n}}$ is an (embedded, non-empty, proper) algebraic variety deûned over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ of logarithmic vector fields of $X$ is a reflexive ${{O}_{{{\mathbb{A}}^{n}}}}$ -module. As a consequence of this result, we derive that if ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ is a free ${{O}_{{{\mathbb{A}}^{n}}}}$ -module, which is shown to be equivalent to the freeness of the $t$ -th exterior power of ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ for some (in fact, any) $t\,\le \,n$ , then necessarily $X$ is a Saito free divisor.

In this paper, we give an explicit realization of the universal SL2-representation rings of free groups by using ‘the ring of component functions’ of SL(2, ℂ)-representations of free groups. We introduce a descending filtration of the ring, and determine the structure of its graded quotients. Then we study the natural action of the automorphism group of a free group on the graded quotients, and introduce a generalized Johnson homomorphism. In the latter part of this paper, we investigate some properties of these homomorphisms from a viewpoint of twisted cohomologies of the automorphism group of a free group.

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

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 describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$ . When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$

A recent result of Eisenbud–Schreyer and Boij–Söderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest ‘wild’ quiver.

Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$ . We call $M$ $\text{H}$ -smaller ( $\text{HS}$ for short) if and only if $M$ is infinite and $\left| M/N \right|\,<\,\,\left| M \right|$ for every nonzero submodule $N$ of $M$ . After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose $M$ is faithful over $R$ , $R$ is a domain (we will show that we can restrict to this case without loss of generality), and $K$ is the quotient field of $R$ . If $M$ is $\text{HS}$ over $R$ , then $R$ is $\text{HS}$ as a module over itself, $R\,\subseteq \,M\,\subseteq \,K$ , and there exists a generating set $S$ for $M$ over $R$ with $\left| S \right|\,<\,\left| R \right|$ . We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

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.

Let R be a commutative ring with non-zero identity and M be a unitary R-module. Let (M) be the set of all submodules of M, and φ: (M) → (M) ∪ {∅} be a function. We say that a proper submodule P of M is a prime submodule relative to φ or φ-prime submodule if a ∈ R and x ∈ M, with ax ∈ P ∖ φ(P) implies that a ∈(P :RM) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ (M), then a φ-prime submodule is exactly a prime submodule. Also if we consider φ(N) = {0} for each submodule N of M, then in this case a φ-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterisations of φ-prime submodules will be given, and we show that under some assumptions prime submodules and φ1-prime submodules coincide.

A proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, implies that either or . In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finitely generated $R$ -module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is finitely generated for all $i\,<\,t$ , then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ is finitely generated. In this paper it is shown that if $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is Artinian for all $i\,<\,t$ , then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ need not be Artinian, but it has a finitely generated submodule $N$ such that $\text{Ho}{{\text{m}}_{R}}\left( R/\mathfrak{a},\text{H}_{\mathfrak{a}}^{t}\left( M \right) \right)/N$ is Artinian.

In this paper we obtain several new characterizations of modules having small cofinite irreducibles. One of these characterizations involves a metric topology defined on the submodule lattice.