Algebra and its Applications

The study of various types of hulls of a module has been of interest for a long time. Our focus in this paper is to present results on some classes of these hulls of modules, their examples, counter examples, constructions and their applications. Since the notion of hulls and its study were motivated by that of an injective hull, we begin with a detailed discussion on classes of module hulls which satisfy certain properties generalizing the notion of injectivity. Closely linked to these generalizations of injectivity, are the notions of a Baer ring and a Baer module. The study of Baer ring hulls or Baer module hulls has remained elusive in view of the underlying difficulties involved. Our main focus is to exhibit the latest results on existence, constructions, examples and applications of Baer module hulls obtained by Park and Rizvi. In particular, we show the existence and explicit description of the Baer module hull of a module N over a Dedekind domain R such that N / t(N) is finitely generated and \(\text {Ann}_R(t(N))\ne 0\), where t(N) is the torsion submodule of N. When N / t(N) is not finitely generated, it is shown that N may not have a Baer module hull. Among applications, our results yield that a finitely generated module N over a Dedekind domain is Baer if and only if N is semisimple or torsion-free. We explicitly describe the Baer module hull of the direct sum of \(\mathbb {Z}\) with \(\mathbb {Z}_p\) (p a prime integer) and extend this to a more general construction of Baer module hulls over any commutative PID. We show that the Baer hull of a direct sum of two modules is not necessarily isomorphic to the direct sum of the Baer hulls of the modules, even if each relevant Baer module hull exists. A number of examples and applications of various classes of hulls are included.

A semigroup is said to be an internal spined product of its subsemigroups if it is naturally isomorphic to an external spined product of the subsemigroups. We shall show that internal spined products can be identified with external spined products in the class of orthocryptogroups. On the other hand, two concepts are not equivalent in general as we give examples of external spined products that admit no internal spined product decomposition. Further, we examine internal spined product of orthocryptogroups. Using a lattice theoretic method, we obtain a unique decomposition theorem similar to the Krull–Schmidt theorem in group theory. We also study completely reducible orthocryptogroups in which any normal sub-orthocryptogroup is a spined factor. We show that such an orthocryptogroup is an internal spined product of simple sub-orthocryptogroups.

Let R be a prime ring of characteristic different from 2, \(Q_r\) its right Martindale quotient ring and C its extended centroid. Suppose that F is a nonzero generalized skew derivation of R, with the associated automorphism \(\alpha \), and \(p(x_1,\ldots ,x_n)\) a noncentral polynomial over C, such that for all \(x,y \in \{p(r_1,\ldots ,r_n) : r_1,\ldots ,r_n \in R\}\). Then \(\alpha \) is the identity map on R and F is an ordinary derivation of R.

This article presents a brief survey of the work done on various additive representations of elements in rings. In particular, we study rings where each element is a sum of units; rings where each element is a sum of idempotents; rings where each element is a sum of idempotents and units; and rings where each element is a sum of additive commutators. We have also included a number of open problems in this survey to generate further interest among readers in this topic.

Let R be a prime ring with center Z(R), J a nonzero left ideal, \(\alpha \) an automorphism of R and R admits a generalized \((\alpha ,\alpha )\)-derivation F associated with a nonzero \({(}\alpha ,\alpha {)}\)-derivation d such that \(d(Z(R))\ne (0)\). In the present paper, we prove that if any one of the following holds: \(\textit{(i)}\) \(F([x,y])-\alpha ([x,y])\in Z(R)\) (ii) \(F([x,y])+\alpha ([x,y])\in Z(R)\) (iii) \(F(x \circ y)-\alpha (x \circ y)\in Z(R)\) (iv) \(F(x \circ y)-\alpha (x \circ y)\in Z(R)\) for all \(x,y\in J\), then R is commutative. Also some related results have been obtained.

Let R be a prime ring with Utumi quotient ring U. If R admits a generalized derivation F associated with a derivation d such that \( F([x^my, x]_k)^n-[x^my, x]_k =0\) for all \( x, y\in R \) where \( m\ge 0\) and \( n, k \ge 1\) fixed integers, then R is commutative or \( n=1 \), \(d=0\) and F is an identity map. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to noncommutative Banach algebras.

We describe recent developments in the study of unimodular rows over a commutative ring by studying the associated group \(SUm_r(R)\), generated by Suslin matrices associated to a pair of rows v, w with \(\langle v, w \rangle = 1\). We also sketch some futuristic developments which we expect on how this association will help to solve a long standing conjecture of Bass–Suslin (initially in the metastable range, and later the entire expectation) regarding the completion of unimodular polynomial rows over a local ring, as well as how this study will lead to understanding the geometry and physics of the orbit space of unimodular rows under the action of the elementary subgroup.

In this expository paper, we present proofs of Grothendieck–Serre formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh’s conjecture by Kummini and Masuti. We present an alternate proof of Huneke–Ooishi Theorem and a generalisation for multi-graded filtrations.

Let K be a field of characteristic zero and let \(\mathcal {O}_n\) be the ring \(K[[X_1,\ldots ,X_n]]\). Let \(\mathcal {D}_n = \mathcal {O}_n[\partial _1,\ldots ,\partial _n]\) be the ring of K-linear differential operators on \(\mathcal {O}_n\). Let M be a holonomic \(\mathcal {D}_n\)-module. In this paper we prove \(H^i({\partial }, M) = 0\) for \(i < n - \dim M\). Here \(\dim M = \) dimension of support of M as an \(\mathcal {O}_n\)-module. Also let \(R = K[X_1,\ldots ,X_n]\) and let I be an ideal in R and let \(A_n(K) = K<X_1,\ldots ,X_n, \partial _1, \ldots , \partial _n>\) be the nth Weyl algebra over K. By a result due to Lyubeznik the local cohomology modules \(H^i_I(R)\) are holonomic \(A_n(K)\)-modules for each \(i \ge 0\). In this article we also compute the de Rham cohomology modules \(H^*(\partial _1,\ldots ,\partial _n ; H^*_I(R))\) for certain classes of ideals.

In 1904, Issai Schur proved the following result. If G is an arbitrary group such that \(G/{\text {Z}} (G)\) is finite, where \({\text {Z}} (G)\) denotes the center of the group G, then the commutator subgroup of G is finite. A partial converse of this result was proved by B.H. Neumann in 1951. He proved that if G is a finitely generated group with finite commutator subgroup, then \(G/{\text {Z}} (G)\) is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups G such that \(|G/{\text {Z}} (G)| = |\gamma _2(G)|^d\), where d denotes the number of elements in a minimal generating set for \(G/{\text {Z}} (G)\). Some problems and questions are posed in the sequel.

In this paper, we develop a Robinson–Schensted algorithm for the walled Brauer algebras which gives the bijection between the walled Brauer diagram d and the pairs of standard tri-tableaux of shape \(\lambda =(\lambda _1,\lambda _2,\lambda _3)\) with \(\lambda _1=(2^{f}),\lambda _2 \vdash r-f\) and \(\lambda _3 \vdash s-f,\) for \(0 \le f \le \min (r,s).\) As a biproduct, we define a Robinson–Schensted correspondence for the walled signed Brauer algebras which gives the correspondence between the walled signed Brauer diagram d and the pairs of standard signed-tri-tableaux of shape \(\lambda =(\lambda _1,\lambda _2,\lambda _3)\) with \(\lambda _1=(2^{2f}),\lambda _2 \vdash _b r-f\) and \(\lambda _3 \vdash _b s-f,\) for \(0 \le f \le \min (r,s).\) We also derive the Knuth relations and the determinantal formula for the walled Brauer and the walled signed Brauer algebras by using the Robinson–Schensted correspondence.

The concept of \(\Gamma \)-semigroup is a generalization of semigroup. Let S and \(\Gamma \) be two nonempty sets. S is called \(\Gamma \)-semigroup if there exists a mapping \(S\times \Gamma \times S\longrightarrow S\), written as \((a, \alpha , b)\longrightarrow a\alpha b\), satisfying the identity \( (a\alpha b)\beta c\) \(=\) \(a\alpha (b\beta c) \) for all \(a, b, c\in S\) and \( \alpha , \beta \in \Gamma \). This article is a survey of some works published by different authors on \(\Gamma \)-semigroups.

In this article, a Schr\({\ddot{\mathrm{o}}}\)der–Bernstein type theorem is proved for orthomodular lattices. Various comparability axioms available in Baer \(*\)-rings are introduced in orthomodular lattices. Some applications to complete orthomodular lattices are given. The related classical results in Baer \(*\)-rings are generalized to \(*\)-rings.

Structure theory of regular semigroups has been using theory of categories to a great extent. Structure theory of regular semigroups developed by K.S.S. Nambooripad using inductive groupoids, structure of combinatorial regular semigroups developed by A.R. Rajan and several other structure theories have made extensive use of categories. The theory of cross connections developed by K.S.S. Nambooripad has provided an abstract description of the category of left ideals of a regular semigroup which he called normal category. The first appearance of categories in structure theory can be traced to Schein’s structure theory of inverse semigroups which uses groupoids as a basic object where groupoids are categories in which all morphisms are isomorphisms. Schein described the category of isomorphisms between order ideals of the set of idempotents of an inverse semigroup and called them inductive groupoids. Some instances of appearance of categories in structure theory of certain classes of regular semigroups are presented here.

In [4] (Structure of regular semigroups, 1979) K.S.S. Nambooripad introduced biordered sets as a partial algebra \((E, \omega ^r, \omega ^l)\) where \(\omega ^r\) and \(\omega ^l\) are two quasiorders on the set E satisfying biorder axioms; to study the structure of a regular semigroup. John von Neumann (Continuous Geometry, 1960 in [5]) described the complemented modular lattice of principle ideals of a regular ring. In this paper, we introduced the biorder ideals of a regular ring and showed that these ideals form a complemented modular lattice.

Let N be a near ring. An additive mapping \(F:N\longrightarrow N\) is said to be a generalized semiderivation on N if there exists a semiderivation \(d:N\longrightarrow N\) associated with a function \(g:N\longrightarrow N\) such that \(F(xy)=F(x)y+g(x)d(y)=d(x)g(y)+xF(y)\) and \(F(g(x))=g(F(x))\) for all \(x,y \in N\). The purpose of the present paper is to prove some theorems in the setting of semigroup ideal of a 3-prime near ring admitting a pair of suitably-constrained generalized semiderivations, thereby extending some known results on derivations and generalized derivations. We show that if N is 2-torsion free and \(F_1\) and \(F_2\) are generalized semiderivations such that \(F_1F_2=0\), then \(F_1=0\) or \(F_2=0\); we prove other theorems asserting triviality of \(F_1\) or \(F_2\); and we also prove some commutativity theorems.

In this paper, we introduce and study the notions of n-strongly Gorenstein projective and injective complexes, which are generalizations of n-strongly Gorenstein projective and injective modules, respectively. Further, we characterize the so-called notions and prove that the Gorenstein projective (resp., injective) complexes are direct summands of n-strongly Gorenstein projective (resp., injective) complexes. Also, we discuss the relationships between n-strongly Gorenstein injective and n-strongly Gorenstein flat complexes, and for any two positive integers n and m, we exhibit the relationships between n-strongly Gorenstein projective (resp., injective) and m-strongly Gorenstein projective (resp., injective) complexes.

Let R be a prime ring with Utumi quotient ring U and extended centroid C, F a nonzero generalized derivation of R, I a nonzero right ideal of R, \(f(r_1,\ldots ,r_n)\) a multilinear polynomial over C and \(s\ge 1, t\ge 1\) be fixed integers. If \((F(f(r_1,\ldots ,r_n))^s-f(r_1,\ldots ,r_n)^s)^t=0\) for all \(r_1,\ldots ,r_n\in I\), then one of the following holds:

In this paper, we have studied the properties of semi-projective module and its endomorphism rings related with Hopfian, co-Hopfian, and directly finite modules. We have provide an example of module which are semi-projective but not quasi-projective. We also prove that for semi-projective module M with \(dim M <\infty \) or \(Codim M < \infty \), \(M^n\) is Hopfian for every integer \(n \ge 1\). Apart from this we have studied the properties of pseudo-semi-injective module and observed that for pseudo-semi-injective module, co-Hopficity weakly co-Hopficity and directly finiteness are equivalent. Finally proved that for pseudo-semi-injective module M, N be fully invariant M-cyclic submodule of M with N is essential in M, then N is weakly co-Hopfian if and only if M is weakly co-Hopfian.

We prove that distinguishing number \(D_{\overrightarrow{{S}_{n}}}(X)\) can be at most \(n+1+[\frac{n}{6}]\) for \(n\le 36\) and find the complete sets of distinguishing numbers \(D_{\overrightarrow{{S}_{2}}}(X)\) and \( D_{\overrightarrow{{A}_{2}}}(X)\). The distinguishing numbers of the actions of \(\overrightarrow{{S}_{3}}\) and \(\overrightarrow{{A}_{3}}\) are also discussed.

In this paper, we examine the preservation of diameter and girth of the zero-divisor graph under extension to Laurent polynomial and Laurent power series rings.

Let R be a prime ring and \(F, G: R\rightarrow R\) be two generalized derivations of R such that \(F^2+G\) is n-commuting or n-skew-commuting on a nonzero square closed Lie ideal U of R. In the present paper we prove under certain conditions that \(U\subseteq Z(R)\).

Zero-divisor graphs and total graphs are most popular graph constructions from commutative rings. Through these constructions, the interplay between algebraic structures and graphs are studied. Indeed, it is worthwhile to relate algebraic properties of commutative rings to the combinatorial properties of assigned graphs. The concept of dominating sets and domination parameters is very important in graph theory due to varied applications. Several authors extensively studied about domination parameters for zero-divisor graphs and total graphs from commutative rings. In this survey article, we present results obtained with regard to domination for zero-divisor graphs and total graphs from commutative rings.

Beaumont studied groups with isomorphic proper subgroups (see Beaumont: Bull Amer Math Soc 51, 381–387 1945 [1]). In Beaumont et al.: Trans Amer Math Soc 91(2), 209–219 1959 [2], Beaumont and Pierce consider the problem of determining all R-modules M over a principal ideal domain R which have proper isomorphic submodules. Such modules are called I-modules. In Chaturvedi: Iso-retractable Modules and Rings (to appear) [3], we investigate iso-retractable modules that is the modules which are isomorphic to their nonzero submodules. Also, a ring R is said to be iso-retractable if \(R_R\) is an iso-retractable module. The class of iso-retractable modules lies in between simple modules and the uniform modules. In the present paper, our main objective is to investigate general properties of iso-retractable modules and rings. Finally, we show that being iso-retractable is a Morita invariant property.

In this paper, we characterize the normal categories associated with a completely simple semigroup \(S = \mathscr {M}[G;I,\Lambda ;P]\) and show that the semigroup of normal cones \(T\mathcal {L}(S)\) is isomorphic to the semi-direct product \(G^\Lambda \ltimes \Lambda \). We characterize the principal cones in this category and the Green’s relations in \(T\mathcal {L}(S)\).

In the present paper, the notions of \((\in , \in \vee q_{k})\)-intuitionistics fuzzy ideal, \((\in , \in \vee q_{k})\)-intuitionistics fuzzy bi-ideal, and \((\in , \in \vee q_{k})\)-intuitionistics fuzzy generalized bi-ideal of an ordered semigroup are introduced. Then, we characterize these various intuitionistic fuzzy ideals. Finally, using the properties of these intuitionistic fuzzy ideals, we have characterized different classes of ordered semigroups.

M. Satyanarayana posed a problem in 2001: which infinite codes are recognizable. Recognizable code means a code accepted by a finite automaton. Equivalently a code X is recognizable iff \(u^{-1}X\) is finite for \(u\in A^{*}\). It is well known that finite codes are recognizable. We partially answer the problem of Satyanarayana. We know that a right complete semaphore suffix code is recognizable. We study here recognizablity of right complete semaphore codes with conditions other than suffix and further dropping semaphore condition also. Barring right completeness, the problem for general infinite codes is still open.