A Note on Induced Modules

1961 ◽  
Vol 13 ◽  
pp. 587-592
Author(s):  
Charles W. Curtis
Keyword(s):  
Identity Element ◽  
Identity Operator ◽  
Finitely Generated ◽  
Standard Construction ◽  
Left And Right

In this paper, A denotes a ring with an identity element 1, and B a subring of A containing 1 such that B satisfies the left and right minimum conditions, and A is a finitely generated left and right B-module. The identity element 1 is required to act as the identity operator on all modules which we shall consider. For any left B-module V, there is a standard construction of a left A -module which is, roughly speaking, the smallest A -module containing V.

scholarly journals On the Class Number of Representations of an Order

1959 ◽  
Vol 11 ◽  
pp. 660-672 ◽  
Author(s):  
Irving Reiner
Keyword(s):  
Prime Ideal ◽  
Class Number ◽  
Identity Element ◽  
Multiplicative Group ◽  
Identity Operator ◽  
List Type ◽  
Finitely Generated ◽  
Quotient Field ◽  
Finite Dimensional ◽  
Torsion Free

We shall use the following notation throughout:R = Dedekind ring (5).u = multiplicative group of units in R.h = class number of R.K = quotient field of R.p = prime ideal in R.Rp = ring of p-adic integers in K.We assume that h is finite, and that for each prime ideal p, the index (R:p) is finite.Let A be a finite-dimensional separable algebra over K, with an identity element e (4, p. 115). Let G be an R-ordev in A, that is, G is a subring of A satisfying(i)e ∈ G,(ii)G contains a i∈-basis of A,(iii)G is a finitely-generated i?-module.By a G-module we shall mean a left G-module which is a finitely-generated torsion-free i∈-module, on which e acts as identity operator.

scholarly journals Adequate Semigroups

1979 ◽  
Vol 22 (2) ◽  
pp. 113-125 ◽  
Author(s):  
John Fountain
Keyword(s):  
Identity Element ◽  
Adequate Semigroup ◽  
Green’S Relation ◽  
Left And Right ◽  
Special Classes

A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.

Finite presentability of generalized Bruck–Reilly ∗-extension of groups

2016 ◽  
Vol 09 (04) ◽  
pp. 1650090 ◽  
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Finite Subset ◽  
Identity Element ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Finite Presentability ◽  
Finitely Presented

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra 242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, [Formula: see text], to be finitely generated and finitely presented. Let [Formula: see text] be a group, [Formula: see text] be morphisms and [Formula: see text] ([Formula: see text] and [Formula: see text] are the [Formula: see text]- and [Formula: see text]-classes, respectively, contains the identity element [Formula: see text] of [Formula: see text]). We prove that [Formula: see text] is finitely generated if and only if there exists a finite subset [Formula: see text] such that [Formula: see text] is generated by [Formula: see text]. We also prove that [Formula: see text] is finitely presented if and only if [Formula: see text] is presented by [Formula: see text], where [Formula: see text] is a finite set and [Formula: see text] [Formula: see text] for some finite set of relations [Formula: see text].

scholarly journals Acyclic complexes of finitely generated free modules over local rings

2009 ◽  
Vol 105 (1) ◽  
pp. 85 ◽  
Author(s):  
Meri T. Hughes ◽  
David A. Jorgensen ◽  
Liana M. Sega
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Reflexive Module ◽  
Commutative Local Ring ◽  
Standard Construction ◽  
Acyclic Complexes ◽  
Free Modules

We consider the question of how minimal acyclic complexes of finitely generated free modules arise over a commutative local ring. A standard construction gives that every totally reflexive module yields such a complex. We show that for certain rings this construction is essentially the only method of obtaining such complexes. We also give examples of rings which admit minimal acyclic complexes of finitely generated free modules which cannot be obtained by means of this construction.

scholarly journals Integer Semigroups Associated with Dumont-Thomas Numeration Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Víctor F. Sirvent
Keyword(s):  
Identity Element ◽  
Finitely Generated ◽  
Numeration System ◽  
Primitive Substitution ◽  
Binary Operations ◽  
Numeration Systems ◽  
Positive Integers

Given a primitive substitution, we define different binary operations on infinite subsets of the nonnegative integers. These binary operations are defined with the help of the Dumont-Thomas numeration system; that is, a numeration system associated with the substitution. We give conditions for these semigroups to have an identity element. We show that they are not finitely generated. These semigroups define actions on the set of positive integers. We describe the orbits of these actions. We also estimate the density of these sets as subsets of the positive integers.

Quasi-automatic semigroups

2020 ◽  
pp. 2150046
Author(s):  
Yanhui Wang ◽  
Yuhan Wang ◽  
Xueming Ren ◽  
Kar Ping Shum
Keyword(s):  
Cayley Graph ◽  
Identity Element ◽  
Zero Element ◽  
The Other ◽  
Finitely Generated ◽  
Converse Statement ◽  
Free Products ◽  
Generating Set ◽  
Automatic Semigroup ◽  
Automatic Group

Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup [Formula: see text] may be automatic only when [Formula: see text] is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with a zero element adjoined is graph automatic, and also a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example 3.6). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] is a subsemigroup with finite Rees index in [Formula: see text] will be investigated and considered.

scholarly journals On product-one sequences over dihedral groups

2020 ◽  
pp. 2250064
Author(s):  
Alfred Geroldinger ◽  
David J. Grynkiewicz ◽  
Jun Seok Oh ◽  
Qinghai Zhong
Keyword(s):  
Finite Group ◽  
Identity Element ◽  
Finite Sequence ◽  
Finitely Generated ◽  
Dihedral Groups ◽  
Group A ◽  
Arithmetic Combinatorics ◽  
A Finite Group ◽  
Extremal Properties

Let [Formula: see text] be a finite group. A sequence over [Formula: see text] means a finite sequence of terms from [Formula: see text], where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over [Formula: see text] (with the concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This paper provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

scholarly journals On diagonal acts of monoids

2001 ◽  
Vol 63 (1) ◽  
pp. 167-175 ◽  
Author(s):  
E. F. Robertson ◽  
N. Ruškuc ◽  
M. R. Thomson
Keyword(s):  
Finitely Generated ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Left And Right ◽  
Finitely Presented

It is proved that the monoid RN of all partial recursive functions of one variable is finitely generated, and that RN × RNis a cyclic (left and right) RN-act (under the natural diagonal actions s (a, b) = (sa, sb), (a, b) s = (as, bs)). We also construct a finitely presented monoid S such that S × S is a cyclic left and right S-act, and study further interesting properties of diagonal acts and their relationship with power monoids.

The Artin–Rees equations

1987 ◽  
Vol 102 (3) ◽  
pp. 385-387
Author(s):  
A. Caruth
Keyword(s):  
Prime Ideal ◽  
Unique Solution ◽  
Noetherian Ring ◽  
Identity Element ◽  
Solution Set ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Maximal Force ◽  
The Family ◽  
Image Position

Let R denote a commutative Noetherian ring with an identity element and N a finitely generated R -module. When K is a submodule of N and A an ideal of R the Artin–Rees lemma states that there is an integer q ≥ 0 such that AnN ∩ K = An−q(AqN ∩ K) for all n ≥ q (Rees[4]; Northcott [3], theorem 20, p. 210; Atiyah and Macdonald [1], proposition 10·9, p. 107; Nagata [2], theorem (3·7), p. 9). The above equation belongs to the family of module equations involving A and K which is considered below. We characterize, in terms of A and K, the set of submodules X of N for which there is an integer q = q(X) ≥ 0 satisfying the equationEquation (1), which we call the Artin–Rees equation related to A and K, gets its maximal force when X is largest and we determine the best possible solution in this sense. Notice that for any submodule X satisfying (1), X ⊆ K:NAn for all n ≥ q(X). Since N is a Noetherian R-module ([3], proposition 1 (corollary), p. 177), there is an integer t ≥ 1 such that K:NAt = K:NAt+n for all n ≥ 0. We define M = K:NAt and prove, in Theorem 1, that X = Q satisfies equation (1), for a suitable integer q(Q) ≥ 0, if and only if K ⊆ Q:NAυ ⊆ M for some integer υ ≥ 0. In topological terms, the A-adic topology of K coincides with the topology induced by the A-adic topology of N on the subspace Q if the inequality K ⊆ Q:NAυ ⊆ M is satisfied. It follows that the solution set of equation (1) includes every submodule of N of the form An−rK when n ≥ r = q(K) as well as every submodule lying between K and M. Hence, X = M is the strongest solution, in the sense that M is the largest such submodule contained in An−s (AsN ∩ K): NAn for all n ≥ s = q(M). Recall that M is strictly larger than K if and only if A is contained in at least one prime ideal of R belonging to K ([3], theorem 14 (corollary 1), p. 193). Thus, equation (1) has a unique solution (necessarily X = K) if and only if A is not contained in any prime ideal of R belonging to any solution.

scholarly journals ITERATED FUNCTION SYSTEMS, REPRESENTATIONS, AND HILBERT SPACE

2004 ◽  
Vol 15 (08) ◽  
pp. 813-832 ◽  
Author(s):  
PALLE E. T. JORGENSEN
Keyword(s):  
Hilbert Space ◽  
Borel Measure ◽  
Equilibrium Measure ◽  
Borel Function ◽  
Iterated Function Systems ◽  
Identity Operator ◽  
Positive Borel Measure ◽  
Iterated Function ◽  
Standard Construction ◽  
Natural Equilibrium

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τi is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τi to operators on the Hilbert space L2(μ). Instead, we show that it is possible to realize the maps τi quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫X|φ|2 dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and [Formula: see text] We prove that, under general conditions on the system (X,τi), there are isometries [Formula: see text] in ℋ(X) satisfying [Formula: see text] the identity operator in ℋ(X). For the construction we assume that some mapping σ:X→X satisfies the conditions σ◦τi= id X, i=1,…,N. We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

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