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of the paper: "G. Picavet and M. Picavet-L'Hermitte, Modules with finitely many submodules, Int. Electron. J. Algebra, 19 (2016), 119-131.":We characterize ring extensions $R \subset S$ having FCP (FIP), where $S$ is the idealization of some $R$-module. As a by-product we exhibit characterizations of the modules that have finitely many submodules. Our tools are minimal ring morphisms, while Artinian conditions on rings are ubiquitous. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M . For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two.

A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

We prove that every curve in the biliaison class of an extremal curve in P3 can be connected to an extremal curve.

We fill a gap in the proof of one of the central theorems in Epp's paper, concerning $p$-cyclic extensions of complete discrete valuation rings., Comment: 3 pages

In section 8.3 of our paper "Duality and Flat Base Change on Formal Schemes" (http://arXiv.org/abs/alg-geom/9708006) some important results concerning localization of, and preservation of coherence by, basic duality functors, were based on the false statement that any closed formal subscheme of an open subscheme of the completion P of a relative projective space is an open subscheme of a closed formal subscheme of P. In this note, the said results are provided with solid foundations.

The idea of residuation goes back to Dedekind, who introduced it into ideal theory. Residuation now plays an important role in several fields of mathematics, especially commutative ring theory. Let R be a commutative ring with identity. The residual of an ideal B with respect to an ideal A is the ideal A: B = {x ∈ R | xB ⊆ A. Here A: B is the largest ideal X of R with the property that BX ⊆ A. Residuation may be defined in other algebraic structures and may be defined independently of multiplication.

The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.

Let I be the Stanley–Reisner ideal of a simplicial complex Δ. In this short paper, we shall give a formula of vanishing of the local cohomology modules for S/I(r) in the case Δ is a union of disjoint matroids, where I(r) is the rth symbolic power of I. As an application, we will improve a previous result in Minh and Nakamura (Nagoya Math. J. 213, 127–140, 2014) for the k-Buchsbaumness of S/I(r).

In this short paper, we show that properties of an ideal I of a ring R are related to those of the homogeneous ideal I(+)IM of a ring R(+)M.

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that these examples are dual to modules over polynomial rings that yield counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules., 7 pages - Complete rewrite of previous draft with new section and shorter proofs

The purpose of this short paper is to give a structure theorem for certain Gorenstein ideals.

In this short paper, our main result gives and internal characterization in A of when the smash products of graded rings have the ideal intersection property. This result generalizes S. Montgomery and D. Passman’s result on criteria for the smash products of graded rings to be prime or simple.

In this short paper I describe special situations where a local ring is normally flat along a divisorial prime ideal.

A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda and Migliore-Nagel concerning the $h$-vectors of quadratic Gorenstein rings., Comment: v2 - Minor changes based on referee comments

In the field of many valued logic, lattice valued logic (especially ideals) plays an important role. Nowadays, lattice valued logic is becoming a research area. Researchers introduced weak LI-ideals of lattice implication algebra. Furthermore, other scholars researched LI-ideals of implicative almost distributive lattice. Therefore, the target of this paper was to investigate new development on the extension of LI-ideal theories and properties in implicative almost distributive lattice. So, in this paper, the notion of weak LI-ideals and maximal weak LI- ideals of implicative almost distributive lattice are defined. The properties of weak LI- ideals in implicative almost distributive lattice are studied and several characterizations of weak LI-ideals are given. Relationship between weak LI-ideals and weak filters are explored. Hence, the extension properties of weak LI-ideal of lattice implication algebra to that of weak LI-ideal of implicative almost distributive lattice were shown.

The mp-quantales were introduced in a previous paper as an abstraction of the lattices of ideals in mp-rings and the lattices of ideals in conormal lattices. Several properties of m-rings and conormal lattices were generalized to mp-quantales. In this paper we shall prove new characterization theorems for mp-quantales and for semiprime mp-quantales (these last structures coincide with the P F-quantales). Some proofs reflect the way in which the reticulation functor (from coherent quantales to bounded distributive lattices) allows us to export some properties from conormal lattices to mp-quantales.

Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Г-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Г-semigroups. Many results in ideal theory of Г-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Г-semigroups. In Section 2, we define almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups by using the concept ideas of interior Г-ideals and almost Г-ideals of Г-semigroups. Every almost interior Г-ideal of a Г-semigroup S is clearly a weakly almost interior Г-ideal of S but the converse is not true in general. The notions of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups are generalizations of the notion of interior Г-ideal of a Г-semigroup S. We investigate basic properties of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Г-semigroups. In Section 3, we introduce fuzzy almost interior Г-ideals and fuzzy weakly almost interior Г-ideals of Г-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Г-ideals [weakly almost interior Г-ideals] and fuzzy almost interior Г-ideals [fuzzy weakly almost interior Г-ideals] of Г-semigroups.

This paper studies the concept of the -quasi-*-IFP (resp., -*-reflexive) *-rings, as a generalization of the quasi-*-IFP (resp., *-reflexive) *-rings and every quasi-*-IFP (resp., *-reflexive) *-ring is -quasi-*-IFP (resp., -*-reflexive). This paper also discusses the sufficient condition for the quasi-*-IFP (resp., *-reflexive) *-ring in order to be -quasi-*-IFP (resp., -*- reflexive). Finally, this study investigates the -quasi-*-IFP (resp., -*-reflexivity) by using some types of the polynomial rings.

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages

In this paper, the traditional proof of “square root of 2 is not a rational number” has been reviewed, and then the theory has been generalized to “if n is not a square, square root of n is not a rational number”. And then some conceptions of ring, integral domain, ideal, quotient ring in Advanced algebra, have been introduced. Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain. Evens and odds are principal ideals in integral domain. The operations on evens and odds are operations on quotient ring. After introducing “the minimalist form” in fraction ring. The paper proves the main conclusion: in a integral domain, multiplicative subset S produces a fraction ring S−1R, and n is not a square element in R, then to every element a∈R, a2≠n.

Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right $H$-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation $$\Delta \Delta u + {{\Bbbk }^{2}}u = f$$ , where the coefficient $$\Bbbk \geqslant 0$$ is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to $$\Bbbk \in [0,{\text{c}}{{{\text{h}}}^{{ - 2}}}]$$ , $$c = {\text{const}}$$ , and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for $$\Bbbk \equiv {\text{const}}{\text{.}}$$ The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.

The notion of a \verb"quasi-prime ideal", for the first time, was introduced in differential commutative rings, i.e. commutative rings considered together with a derivation, as differential ideals maximal among those not meeting some multiplicatively closed subset of a ring. The notion of a semiring derivation is traditionally defined as an additive map satisfying the Leibnitz rule. Due to rapid development of semiring theory in recent years, the need of considering ideals in semirings defined by similar conditions arose. The present paper is devoted to investigating the notion of a \verb"quasi-prime ideal" of differential semiring (which is defined as a semiring together with a derivation on it), not necessarily commutative. It aims to show, how \verb"quasi-prime ideals" are related to prime differential ideals, primary ideals, maximal ideals and other types of ideals of semirings. The paper consists of two main parts. In the first part, the author investigates some properties of quasi-prime differential ideals, and gives some examples of such semiring ideals, such as prime differential, maximal differential ideals, or ideal obtained by derivation operator acting on a prime ideal of a semiring. It contains a theorem, which gives equivalent conditions for a quasi-prime semiring ideal to be prime. The second part of the paper is devoted to considering chains of \verb"quasi-prime ideals". In this part, the interrelation between \verb"quasi-prime ideals" and other types of differential ideals of semirings is established. It contains a theorem, which gives a characterization of such ideals in case of a commutative semiring. This characterization involves the notion of the radical of an ideal of a semiring and a derivation operator for semirings. The paper ends with a theorem, which states that every chain of quasi-prime ideals of a semiring has the least upper bound and the greatest lower bound. It is also proven that every \verb"quasi-prime ideal" containing some differential ideal contains a \verb"quasi-prime ideal" minimal among all the quasi-prime ideals of the given semiring, which contain the above mentioned differential ideal.

This paper presents a new ring consisting of adjacency matrices of graphs. New definitions of binary sum and product of adjacency matrices are being introduced. The paper shows the basic properties for a ring are being satisfied.

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

A signed graph is a pair $$(G,\tau )$$ of a graph G and its sign $$\tau $$ , where a sign $$\tau $$ is a function from $$\{ (e,v)\mid e\in E(G),v\in V(G), v\in e\}$$ to $$\{1,-1\}$$ . Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal $$I_{(G,\tau )}$$ associated with a signed graph $$(G,\tau )$$ , and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of $$I_{(G,\tau )}$$ and then focus on the complete intersection property. More precisely, we find a complete list of graphs G such that $$I_{(G,\tau )}$$ is a complete intersection for every sign $$\tau $$ .

A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke., to appear in Trans. Amer. Math. Soc

In the present paper, the notion of (additive) local multiplier on the ring of matrices over an arbitrary unital associative ring is introduced and investigated. It is proved that every local left multiplier on the matrix ring over a division ring is a left multiplier. Also, in the present paper, the notion of (additive) local Jordan multiplier on a Jordan ring of symmetric matrices is introduced and investigated. It is proved that every local Jordan multiplier on the Jordan ring of symmetric matrices over the field of rational numbers is a Jordan multiplier. Also corollaries of these results are proved.

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

The aim of this paper is to present a Grobner-Shirshov basis for a special type of braid monoids, namely the symmetric inverse monoid In, in terms of the dex-leg ordering on the related elements of monoid. By taking into account the Grobner-Shirshov basis, the ideal form (or, equivalently, the normal form structure) of this important monoid will be obtained. This ideal form will give us the solution of the word problem. At the final part of this paper, we give an application of our main result which find out a Grobner-Shirshov basis for the symmetric inverse monoid I4 such that the accuracy and efficiency of this example can be seen by GBNP package in GAP (Group, Algorithms and Programming) which computes Grobner bases of non-commutative polynomials [ 1 ].

In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $\ast\subset S^{3}\hookrightarrow\ldots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\hookrightarrow\ldots$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomology (over $n$ and $k$, where $k\to\infty$ as $n\to\infty$) of the Massey products constructed. As an application we get that there are non-trivial differentials $d_k$, for arbitrarily large $k$ as $n\to\infty$ in the Eilenberg--Moore spectral sequence connecting the rings $H^*(\Omega X)$ and $H^*(X)$ with coefficients in a field, where $X=\mathcal Z_{P^n}$., Comment: 53 pages, 5 figures; extended version of a paper to appear in Izv. Math

The concept of the annihilating-ideal graph of a commutative ring was introduced by Behboodi et. al in 2011. In this paper, we extend this concept to the hypergraph for which we define an algebraic structure called k-annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. Let R be a commutative ring and k an integer greater than 2 and let A(R,k)be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by AGk(R), is a hypergraph with vertex set A(R,k), and for distinct elements I1,I2,…,Ikin A(R,k), the set {I1,I2,…,Ik}is an edge of AGk(R)if and only if ∏i=1kIi=(0)and the product of any (k−1)elements of the {I1,I2,…,Ik}is nonzero. In this paper, we provide a necessary and sufficient condition for the completeness of 3-annihilating-ideal hypergraph of a commutative ring. Further, we study the planarity of AG3(R)and characterize all commutative ring R whose 3-annihilating-ideal hypergraph AG3(R)is planar. Keywords: Annihilating-ideal, Hypergraph, Incidence graph, Planar, k-annihilating-ideal hypergraph

Обобщено разделение целочисленного секрета, использующего алгоритм китайской теоремы об остатках на случай кольца многочленов от нескольких переменных над конечным полем. Для генерации частичных секретов вместо целочисленных модулей применяются идеалы и их базисы Грёбнера. Этот подход предложен нами ранее. В настоящей работе показано, что любую пороговую структуру доступа можно реализовать идеально. Это является одним из преимуществ предлагаемого подхода. В кольце целых чисел никакую структуру доступа нельзя осуществить идеально, поскольку частичные секреты всех участников имеют различные размеры. This paper deals with a generalization of the secret sharing using Chinese remainder theorem over the integers to multivariate polynomials over a finite field. We work with the ideals and their Gröbner bases instead of integer moduli. Therefore, the proposed method is called GB secret sharing. It was initially presented in our previous paper. Now we prove that any threshold structure has ideal GB realization. In a generic threshold modular scheme in ring of integers the sizes of the share space and the secret space are not equal. So, the scheme is not ideal and our generalization of modular secret sharing to the multivariate polynomial ring is more secure.

The aim of this paper is to introduce and study a new kind of graphs associated to an ideal of a commutative ring. Let ℛ be a commutative ring with identity, and I(ℛ) be the set of all non-trivial ideals of ℛ with S I(ℛ). The sum ideal graph associated to S, denoted by Ψ(ℛ, S), is the undirected graph with vertex set {A I(ℛ): S⊂A+B, for some B I(ℛ)} where two ideal vertices A and B are adjacent if and only if A B and S⊂A+B. In this paper we establish some of characterizations and results of this kind of graph with providing some examples.

In the direction towards the question when mixed multiplicities are equal to the Hilbert-Samuel multiplicity of joint reductions, this paper not only generalizes [28, Theorem 3.1] that covers the Rees's theorem [13, Theorem 2.4], but also removes the hypothesis that joint reductions are systems of parameters in [28, Theorem 3.1]. The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity of joint reductions become closer., 15 pages

A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the $2$-generator property. In the present paper we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains., Comment: This paper has been accepted for publication in the Communications in Algebra

In this paper, we introduce the concept of canonical stretched rings, sparse stretched rings and maximum sparse ideals. Then we give characterizations of canonical stretched rings and sparse stretched rings; and a characterization of Gorenstein rings in terms of their maximum sparse ideals. Several explicit examples are provided along the paper to illustrate such rings., Accepted Acta Mathematica Vietnamica

Let R be an associative ring with identity and M be a left R-module. In this paper, we introduce M-Gorenstein injective modules as a generalization of Gorenstein injective modules. We verify some properties of M-Gorenstein injective modules analogous to those holding for Gorenstein injective modules. There is an interesting theorem in classical homological algebra which asserts that R is a Noetherian ring if and only if the class of injective modules over R is closed under arbitrary direct sum. Our goal in this paper is to investigate the M-Gorenstein injective counterpart of this fact. If the class of M-Gorenstein injective modules over R is closed under arbitrary direct sum, then R will be a Noetherian ring. Also, it has been proved that in the special case M=R, when R is a commutative Noetherian ring with a dualizing complex, then the class of R-Gorenstein injective modules is closed under arbitrary direct sum. In the main theorem of this paper, we prove the general case of this result. More precisely, we show that for any left R-module M over a Noetherian ring R in which every R-module has finite M-Gorenstein injective dimension, the class of M-Gorenstein injective modules is closed under arbitrary direct sum.

In this paper, we study a special class of elements in the finite commutative rings called involutions. An involution of a ring R is an element with the property that x^2-1=0 for some x in R. This study describes both the implementation and enumeration of the involutions of various rings, such as cyclic rings, non-cyclic rings, zero-rings, finite fields, and especially rings of Gaussian integers. The paper begins with simple well-known results of an equation x^2-1=0 over the finite commutative ring R. It provides a concrete setting to enumerate the involutions of the finite cyclic and non-cyclic rings R, along with the isomorphic relation I(R)≅Z_2^k.

For an commutative ring T with identity, we define Distributive ring if any module over T is distributive. In this paper we study and investigate some new relationships between Distributive and hereditary rings. Also, we give many examples and properties of Distributive ring and investigate the behavior of these properties under another ring namely hereditary ring. Also, we also define study these rings and give a characterization of such rings. Some new results have been studied in this paper. Finally, we justify some conditions under which several concepts which are Distributive.

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function $${{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1$$ , where $$I^{(t)}$$ denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and $${{\,\mathrm{pd}\,}}$$ denotes the projective dimension. It has been an open question whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is non-increasing if I is a squarefree monomial ideal. We show that $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is almost non-increasing in the sense that $${{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for all $$s \ge 1$$ and $$t \in E(s)$$ , where $$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$ (which contains all integers $$t \ge (s-1)^2+1$$ ). The range E(s) is the best possible since we can find squarefree monomial ideals I such that $${{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for $$t \not \in E(s)$$ , which gives a negative answer to the above question. Another open question asks whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is always constant for $$t \gg 0$$ . We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that $$I^{(t)}$$ is integrally closed for $$t \gg 0$$ (e.g. if I is a squarefree monomial ideal), then $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is constant for $$t \gg 0$$ with $$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$ Our last result (which is the main contribution of this paper) shows that for any positive numerical function $$\phi (t)$$ which is periodic for $$t \gg 0$$ , there exist a polynomial ring R and a homogeneous ideal I such that $${{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)$$ for all $$t \ge 1$$ . As a consequence, for any non-negative numerical function $$\psi (t)$$ which is periodic for $$t \gg 0$$ , there is a homogeneous ideal I and a number c such that $${{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c$$ for all $$t \ge 1$$ .

The purpose of this paper is to solve the equality-constrained minimization problem of polynomial functions. Let $${\mathbb {R}}$$ be the field of real numbers, and $${\mathbb {R}}[x_1,\ldots ,x_n]$$ the ring of polynomials over $${\mathbb {R}}$$ in variables $$x_1$$ , ..., $$x_n$$ . For an $$f\in {\mathbb {R}}[x_1,\ldots ,x_n]$$ and a finite subset H of $${\mathbb {R}}[x_1,\ldots ,x_n]$$ , denote by $${\mathscr {V}}(f:H)$$ the set $$\{f({\bar{\alpha }})\mid {\bar{\alpha }}\in {\mathbb {R}}^n, \hbox { and }h({\bar{\alpha }})=0,\,\forall h\in H\}$$ . In this paper, we provide some effective algorithms for computing the accurate value of the infimum $$\inf {\mathscr {V}}(f:H)$$ of $${\mathscr {V}}(f:H)$$ , deciding whether or not the constrained infimum $$\inf {\mathscr {V}}(f:H)$$ is attained when $$\inf {\mathscr {V}}(f:H)\ne \pm \infty $$ , and finding a point for the constrained minimum $$\min {\mathscr {V}}(f:H)$$ if $$\inf {\mathscr {V}}(f:H)$$ is attained. With the aid of the computer algebraic system Maple, our algorithms have been compiled into a general program to treat the equality-constrained minimization of polynomial functions with rational coefficients.

Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar., Final version; minor changes made according to the referee report; to appear in the Journal of Algebra

In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict. Many of the rings given in the examples were infinite. In this paper, where possible, examples of minimal finite rings of the various types are given. Along with the rings in the previous taxonomy, NI, abelian and reflexive rings are also included.

In this paper, we introduce a new involutive division, called D-Nœther division, and the corresponding notion of a Nœther basis. It is shown that an ideal is in Nœther position, if and only if it possesses a finite Nœther basis. We present a deterministic algorithm which, given a homogeneous ideal, finds a linear change of variables so that the ideal after performing this change possesses a finite Nœther basis (and equivalently is in Nœther position). Furthermore, we define the new concept of an ideal of Nœther type and study its connections with Rees decompositions. We have implemented all the algorithms described in this paper in Maple and assess their performance on a number of benchmark examples.

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages