Invertibility and Class Number of Orders

1971 ◽  
Vol 23 (1) ◽  
pp. 1-11
Author(s):  
Howard Gorman
Keyword(s):  
Class Number ◽  
Symmetric Algebra ◽  
Preliminary Results ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Valuation Rings ◽  
Work Done ◽  
Equivalent Conditions

This paper continues the work begun in [5] and concerns the invertibility of modules in a finite-dimensional, symmetric algebra L with 1 over a field. In particular, we continue the work done in [5] which dealt with the connection between invertibility in these algebras and a condition called the Brandt Condition, which is a reformulation by Kaplansky [6] of some ideas of Brandt.We begin by proving some preliminary results on invertibility and some equivalent conditions for the dual of an order to be principal.Then, we define the class number of an order and reformulate the concept of invertibility in terms of class number. In this terminology, we find some equivalent conditions which ensure that an order in certain algebras L (including commutative, symmetric algebras, and algebras with a strong involution) has class number equal to 1 (i.e., all modules principal), and we characterize a class of Brandt algebras over the quotient fields of valuation rings as those in which all orders have class number less than or equal to 2.

scholarly journals Silting and Tilting for Weakly Symmetric Algebras

2021 ◽  
Author(s):  
Jenny August ◽  
Alex Dugas
Keyword(s):  
Symmetric Algebra ◽  
The Other ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Other Hand ◽  
Weakly Symmetric

AbstractIf A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in Kb(projA) are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

scholarly journals Valuation rings in finite-dimensional division algebras

1989 ◽  
Vol 120 (1) ◽  
pp. 90-99 ◽  
Author(s):  
H.H Brungs ◽  
Joachim Gräter
Keyword(s):  
Division Algebras ◽  
Finite Dimensional ◽  
Valuation Rings

scholarly journals Integrable Derivations and Stable Equivalences of Morita Type

2018 ◽  
Vol 61 (2) ◽  
pp. 343-362 ◽  
Author(s):  
Markus Linckelmann
Keyword(s):  
Hochschild Cohomology ◽  
Block Theory ◽  
Discrete Valuation Rings ◽  
Symmetric Algebras ◽  
Valuation Rings ◽  
Transfer Maps ◽  
Stable Equivalences ◽  
Morita Type

AbstractUsing that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

SOME PROPERTIES ON ISOCLINISM OF LIE ALGEBRAS AND COVERS

2008 ◽  
Vol 07 (04) ◽  
pp. 507-516 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
HADI BIGDELY ◽  
VAHID ALAMIAN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Direct Sums ◽  
Schur Multipliers ◽  
Finite Dimensional ◽  
Equivalent Conditions

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

scholarly journals A Representation Theoretic Study of Non-Commutative Symmetric Algebras

2019 ◽  
Vol 62 (3) ◽  
pp. 875-887
Author(s):  
D. Chan ◽  
A. Nyman
Keyword(s):  
Structure Theorem ◽  
Symmetric Algebra ◽  
Derived Equivalence ◽  
Division Rings ◽  
Symmetric Algebras

AbstractWe study Van den Bergh's non-commutative symmetric algebra 𝕊nc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that 𝕊nc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1.

Algebra Versus Its Anti-symmetric Algebra

2009 ◽  
Vol 16 (04) ◽  
pp. 661-668
Author(s):  
Seul Hee Choi ◽  
Jongwoo Lee ◽  
Ki-Bong Nam
Keyword(s):  
Lie Algebra ◽  
Symmetric Algebra ◽  
Symmetric Algebras ◽  
Algebra Versus

For a given algebra A= 〈A,+,·〉, we can define its anti-symmetric algebra A-= 〈A-,+,[ , ]〉 using the commutator [ , ] of A, where the sets A and A- are the same. We show that there are isomorphic algebras A1 and A2 such that their anti-symmetric algebras are not isomorphic. We define a special type Lie algebra and show that it is simple.

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.

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