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Let (A , m A) be a Cohen–Macaulay local ring. Let M be a finitely generated A -module and let M ∗ denote the A -dual of M. Furthermore, if M ∗ is a maximal Cohen–Macaulay A -module, then we prove that μ A (M ∗) ≤ μ A (M) e (A) , where μ A (M) is the cardinality of a minimal generating set of M as an A -module and e (A) is the multiplicity of the local ring A. Furthermore, if M is a reflexive A -module then μ A (M) e (A) ≤ μ A (M ∗). As an application, we study the bound on the minimal number of generators of specific modules over two-dimensional normal local rings. We also mention some relevant examples. [ABSTRACT FROM AUTHOR]

For a group G and a finite set A, denote by End (A G) the monoid of all continuous shift commuting self-maps of A G and by Aut (A G) its group of units. We study the minimal cardinality of a generating set, known as the rank, of End (A G) and Aut (A G) . In the first part, when G is a finite group, we give upper and lower bounds for the rank of Aut (A G) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then End (A G) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups. [ABSTRACT FROM AUTHOR]

Let G be a nilpotent group of class two and G ⊗ G be its nonabelian tensor square. In this paper, we determine an upper bound for d ((G ⊗ G) ⊗ G) in terms of d(G), where d(G) is the minimal number of generators of G. In particular, we show that the bound is attained if G is a finitely generated free nilpotent group of class two. [ABSTRACT FROM AUTHOR]

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let I ⊂ K [ x 1 , ... , x n ] be a monomial ideal and let G(I) denote the (unique) minimal monomial generating set of I. How small can | G (I i) | be in terms of | G (I) | ? Until recently, it was widely expected that | G (I 2) | ≥ | G (I) | would always hold. The first counterexamples emerged in 2018 for n = 2. In this article we show that for any n and d there is an m -primary monomial ideal I ⊂ K [ x 1 , ... , x n ] such that | G (I) | > | G (I i) | for all i ≤ d. [ABSTRACT FROM AUTHOR]

Abstract Let I ⊂ K [ x , y ] be a monomial ideal. How small can μ (I 2) be in terms of μ (I) , where μ denotes the least number of generators? It was widely expected that the inequality μ (I 2) > μ (I) should hold whenever μ (I) ≥ 2. Here we disprove this expectation and provide a somewhat surprising answer to the above question. [ABSTRACT FROM AUTHOR]

Abstract: Let p be a prime. We classify finitely generated pro-p groups G which satisfy for all open subgroups H of G. Here denotes the minimal number of topological generators for the subgroup H. Within the category of p-adic analytic pro-p groups, this answers a question of Iwasawa. [Copyright &y& Elsevier]

Abstract: In this paper we characterize the possible Hilbert functions of a local Artinian level ring of given type and socle degree in the codimension two case. In particular, given an “admissible” numerical function h, an effective method is given to produce a zero-dimensional ideal I of such that is a local Artinian level ring having h as Hilbert function. As a consequence we recover the well-known result proved by Briançon and Iarrobino concerning the Hilbert functions of a complete intersection of height two. We extend notions of standard or Gröbner bases from the graded to the local algebra context. The principal tools concern the concepts of enhanced standard basis and generic initial ideals which are defined for an ideal in the formal power series ring . [Copyright &y& Elsevier]