Narrow Results

For rings with involution, we describe the close interrelation between localizations with respect to group inverses and Moore–Penrose inverses, respectively. In full generality, we show that Moore–Penrose localizations are equivalent with certain Fountain–Gould localizations, and that "full" Moore–Penrose and Fountain–Gould orders, respectively, coincide in a perfect or strongly regular ring.

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R_𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R_𝔪. As a corollary, we obtain the computability of the category of finitely presented R_𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

Let R ⊂ S be an extension of integral domains. We say that (R, S) is a well-centered pair of rings, if each intermediate ring T between R and S is well-centered on R, in the sense that each principal ideal of T is generated by an element of R. The aim of this paper is to study well-centered pairs of rings. We investigate the structure of the intermediate rings T between R and S that are well-centered on R. We establish the relationship between well-centered pairs and normal pairs. We present some examples and counterexamples illustrating our theory and showing the limits of our results.

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring $R$ for which idempotents lift modulo the Jacobson radical $J(R)$, but idempotents do not lift modulo $J({𝕄}_2(R))$. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if $I$ and $J$ are ideals of $R$ for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over $I+J$. On the positive side, if $I$ and $J$ are enabling ideals in $R$, then $I+J$ is also an enabling ideal. We show that if $I⊴R$ is (weakly) enabling in $R$, then $I[t]$ is not necessarily (weakly) enabling in $R[t]$ while $I⟦t⟧$ is (weakly) enabling in $R⟦t⟧$. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

We study weight modules of the Lie algebra $W_2$ of vector fields on ${ℂ}^2$. A classification of all simple weight modules of $W_2$ with a uniformly bounded set of weight multiplicities is provided. To achieve this classification, we introduce a new family of generalized tensor $W_n$-modules. Our classification result is an important step in the classification of all simple weight $W_n$-modules with finite weight multiplicities.

As an extension of the class of Dedekind domains, we have introduced and studied the class of multiplicatively pinched-Dedekind domains (MPD domains) and the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains) ([T. Dumitrescu and S. U. Rahman, A class of pinched domains, Bull. Math. Soc. Sci. Math. Roumanie52 (2009) 41–55] and [T. Dumitrescu and S. U. Rahman, A class of pinched domains II, Comm. Algebra39 (2011) 1394–1403]). The main interest of this paper is to study GMPD domains that have only finitely many overrings.

Let $(R,𝔪)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $𝔭$ of $M$ such that depth $M=\mathrm{\text{dim}}R/𝔭$. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen–Macaulay modules with maximal depth are classified. Finally, the attached primes of $H_{𝔪}^i(M)$ are considered for $i<\mathrm{\text{dim}}M$.

We investigate additional properties of protolocalizations, introduced and studied by Borceux, Clementino, Gran, and Sousa, and of protoadditive reflections, introduced and studied by Everaert and Gran. Among other things, we show that there are no non-trivial (protolocalizations and) protoadditive reflections of the category of groups, and establish a connection between protolocalizations and Kurosh–Amitsur radicals of groups with multiple operators whose semisimple classes form subvarieties.

The aim of this paper is to present an algorithm for the primary decomposition of a submodule $N$ of $M\subseteq \mathbb{Q}{[x_1,\dots ,x_n]}^s$. We will use for this purpose the algorithms for primary decomposition for ideals in polynomial rings. We will generalize the method of Kawazoe and Noro to primary decomposition for submodules of free modules.

In this paper, we study the notion of $MV$-modules of fractions with respect to a ⋅-prime ideal $P$. Also the relation between the ideals of $MV$-module $M$ and the ideals of $MV$-module of fractions $M_P$ is studied and it is shown that there is a one to one correspondence between $Q$-prime $A$-ideals of $MV$-module $M$ and $Q_P$-prime $A_P^{\prime }$-ideals of $MV$-module of fractions $M_P$, where $A_P^{\prime }$ is the maximal localization of $A$ at a maximal ⋅-ideal $P$ of $A$ and $Q$ is a ⋅-prime ideal of $A$ such that $Q\subseteq P$.

The aim of the paper is to extend the class of generalized Weyl algebras (GWAs) to a larger class of rings (they are also called GWAs) that are determined by two ring endomorphisms rather than one as in the case of ‘old’ GWAs. A new class of rings, the diskew polynomial rings, is introduced that is closely related to GWAs (they are GWAs under a mild condition). Simplicity criteria are given for GWAs and diskew polynomial rings.

Let $D$ be an integral domain with quotient field $K$, $R$ be an overring of $D$, $X$ be an indeterminate over $R$ and $E$ be a subset of $K$. We consider the ring of $D$-valued$R$-polynomials on$E$, denoted by ${\mathrm{\text{Int}}}_R(E,D)$, formed by the polynomials $f\in R[X]$ such that $f(a)\in D$ for each $a\in E$, that is, ${\mathrm{\text{Int}}}_R(E,D):=\{f\in R[X]:f(E)\subseteq D\}$. In this paper, we study localization properties, local freeness and faithful flatness of ${\mathrm{\text{Int}}}_R(E,D)$ over various classes of locally essential domains. In particular, we extend some known results to more general settings.