Gerhard O. Michler

Structure of semi-perfect hereditary noetherian rings

The object of this paper is to determine the structure of a (not necessari

Die primitiven Klassen arithmetischer Ringe

commutatkc) semi-perfect hereditq noetherian ring. Since a complete semi-local ring is semi-perfect, WC find the structure of a complctc semi-local hereditary noctherian ring as a special case of our main results. .I<ach hcrcditary noctherian ring H is a split extension of a nilpotcnt ring R by a hereditary noetherian semi-prime ring S such that R is a finiteI!; generated pl-ojective S-module (‘l’hcorem 2.2). By a result of L. IAX;;; [:I S is a direct sum of finitely man!? hereditary noetherian prime rings Si ,, If R is semi-perfect, so is each ring S, . Hence it sut%ces to determine the structure of a noetherian hercditarv semi-pcrfcct prime ring, which is given bq

Character correspondences in finite general linear, unitary and symmetric groups

W~ihrend es zahlreiche Untersuchungen fiber primitive Klassen (Variet~iten) von Gruppen gibt, wie zum Beispiel das Buch [7] von Hanna Neumann zeigt, sind die primitiven Klassen yon Ringen in der Literatur kaum behandelt worden. Einen Beitrag hierzu sol1 die vorliegende Note liefern, in der die primitiven Klassen arithmetischer Ringe vollst~indig bestimmt werden. Arithmetische Ringe wurden yon Fuchs in seiner Arbeit [3] als eine Verallgemeinerung der kommutativen Dedekind-Ringe eingefiihrt; dabei heiBt ein (nicht notwendig kommutativer, aber assoziativer) Ring R arithmetisch, wenn der Verband der zweiseitigen Ideale von R distributiv ist. Eine Klasse 91 von Ringen ist primitiv (vgl. Cohn [2], S. 162), wenn 91 abgeschlossen ist beztiglich der Bildung von Unterringen, epimorphen Bildern und direkten (cartesischen) Produkten. Zu einer beliebigen Klasse R von Ringen gibt es stets eine kleinste primitive Klasse, die R enth~lt; sie heiBt die von R erzeugte primitive Klasse. Mit diesen Definitionen gilt der folgende

PRIME RIGHT IDEALS AND RIGHT NOETHERIAN RINGS

Let r > 0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauers and Robinsons solution of Nakayamas conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r > 2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible characters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height h t z = 0 . Therefore Alperins conjecture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspondences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-subgroup /~-~R such that there is a natural height preserving one-to-one correspondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b).

Goldman's primary decomposition and the tertiary decomposition

This chapter discusses the prime right ideals and right noetherian rings. A well-known theorem asserts that a commutative ring R is noetherian if every prime ideal of R is finitely generated. Using the definition of a prime right ideal, it is shown that Cohens theorem holds for any ring. The application of result gives an easy proof for the fact that the power series ring R [ X, α ] of the right noetherian ring R with the surjective endomorphism α is right noetherian. It is found that if every two-sided ideal of R and every prime right ideal of R is a finitely generated right ideal of R , then R is right noetherian. It is found that as every two-sided ideal of R is finitely generated, R satisfies the ascending chain condition on two-sided ideals. It is observed that if R is not right noetherian, then there is an ideal M of R that is maximal among the ideals X of R such that R / X is not right noetherian.

Fourier transforms with respect to monomial representations

Emmy Noether’s theory of the primary decomposition of a submodule of a finitely generated module over a commutative noetherian ring was generalized for modules over a not necessarily commutative left noetherian ring R by Lesieur and Croisot [4], and recently by 0. Goldman [2]. In this note the relations between Goldman’s primary decomposition theory and the tertiary decomposition theory of Lesieur and Croisot are studied. It is shown that each finitely generated Goldman-primary R-module is tertiary (Corollary 2.2). The converse does not hold by Remark 2.3. In fact, each finitely generated tertiary R-module is Goldman-primary if and only if nonisomorphic, indecomposable, injective R-modules have different associated prime ideals (Theorem 2.4). On the other side by Proposition 1.1 there always exists a natural one-to-one correspondence between the set r of all isomorphism classes of indecomposable, injective R-modules and the set p of all prime kernel functors of the category

The Alperin-McKay conjecture holds in the covering groups of symmetric and alternating groups, p ≠2.

01 of all R-modules. As an application of the above results a short proof of the following theorem is given which, because of Theorem 2.4, is equivalent to a theorem of Gabriel [2]: If each finitely generated tertiary R-module over the left noetherian ring R is Goldman-primary, then the Krull-dimension of R in the sense of Gabriel-Rentschler [A coincides with the usual Krull-dimension of R, if it is finite (Theorem 3.4). All rings R considered in this note have an identity element. Each R-module is a unitary left R-module. E(M) always denotes the injective envelope of the R-module M. Concerning the terminology we refer to Lesieur and Croisot [4] and Herstein [3].

Right symbolic powers and classical localization in right Noetherian rings

Let F be a field of characteristic zero, and let f : G --+ F be a function from the finite group G into F. The Fourier transform of f a t the representation p : G ~ GL(m, F) is defined as f ( p ) = ~g~af(g)p(#)~ Mat(m, F), which denotes the ring of all m x m matrices with entries in F. The actual evaluation off(p) can be a tremendous computational task. However, this calculation is required for applications in statistics and elsewhere. The case where p is a permutation or monomial representation is attractive for computation because the sparseness of the matrices p (g) makes direct computation o f f ( p ) relatively easy. On the other hand most applications also require the calculation of the inverse Fourier transform, given by the formula