PUBL1CATIONES MATHEM ATIC AE 24.

KÖTET

D E B R E C E N 1977

ALAPITOTTÄK: RElNYI ALFRED, SZELE TIBOR ES VARGA O T T O

D A R Ö C Z Y Z O L T Ä N , GYIRES BE-LA, R A P C S Ä K A N D R Ä S , T A M Ä S S Y LAJOS KÖZREM ÜKÖDESiVEL SZERKESZTI: BARNA BEI LA

A DEBRECENI TUDOMÄNYEGYETEM

MATEMATIKAI INT^ZETE

INDEX EaüHoe JJ. R . — C a p a t f o e a

F . X p . , 0 6 OAHOM BapHaHTe MeTOßa ycpeAHeHHH RSIX HejiHHeü-

MHTerpoflHixypaBHeHHft craußapTHoro BHßa

205 Behrens, E . - A . , Topologically arithmetical rings of continuous functions 107 B o w m a n , H . — M o o r e , J. D . A method for obtaining proper classes of short exact sequences of Abelian groups 59 C o h n , P . M . , Füll modules over semifirs 305 Daröczy, Z.—Lajkö, K.—Szekelyhidi, L . , Functional equations on ordered fields 173 Dhombres, J. G., Iteration lineaire d'ordre deux 277 D i k s h i t , H . P . , Absolute total-effective Nörlund method 215 E n e r s e n , P . — L e a v i t t , W. G., A note on semisimple classes 311 F r i t z s c h e , R . . Verallgemeinerung eines Satzes von Kulikov, Szele und Kertesz 323 G i l m e r , R . , Modules that are finite sums of simple submodules 5 Glevitzky, B . , O n polynomial regression of polynomial and linear statistics 151 Graef, J. R.—Spikes, P . W., Asymptotic properties of Solutions of a second order nonlinear differential equation 39 Gregor, J., Tridiagonal matrices and functions analytic in two half-planes 11 G u p t a , V. C.—Upadhyay, M . D . , Jntegrability conditions of a structure f statisfying f * - A * f = 0 249 Gyires, B . , On the asymptotic behavoiour of the generalized multinomial distributions 162 Györy, K . , Representation des nombres entiers par des formes binaires 363 Jakäh L . , Ü b e r äquivalente parameterinvariante Variations-probleme erster Ordnung 139 Jensen, C. U., Some remarks on valuations and subfields of given codimension in algebraically closed fields 317 K a r u n a k a r a n , V., A certain radius of convexity problem 1 Kätai, L , The distribution of additive functions on the set of divisors 91 Kätai, /., Research Problems in number theory 263 Knebusch, M . , Remarks oh the paper "Equivalent topological properties of the space of signatures of a semilocal ring" by A . Rosenberg and R. Ware ". 181 K u m a r , V., Convergence of Hermite—Fejer interpolation polynomial on the extended nodes 31 Lajkö, K.—Szekelyhidi. L.—Daröczy, Z . , Functional equations on ordered fields 173 L a l , H . , On radicals in a certain class of semigroups 9 Lambek, J . — M i c h l e r , G., On products of füll linear rings 123 Leavitt, W. G . — E n e r s e n , P . , A note on semisimple classes 311 L o o n s t r a , F . , Subproducts and subdirect products 129 M a k s a , Gy., O n the functional equation f ( x + y ) + g ( x y ) = h ( x ) + h ( y ) 25 Märki, L . — R e d e i , L . , Verallgemeinerter Summenbegriff in der /?-adischen Analysis mit Anwendung auf die endlichen p-Gruppen 101 M i c h l e r , G.—Lambek, J., On products of füll linear rings 123 M l i t z , R . , Kurosch-Amitsur-Radikale in der universalen Algebra 333 M o o r e , J. D . — B o w m a n , H . , A method for obtaining proper classes of short exact sequences of Abelian groups 59 Nagy, B . , A sine functional equation in Banach algebras 77 Nagy, B . , On some functional equations in Banach algebras 257 N g u y e n X u a n Ky, On derivatives of an algebraic polynomial of best approximation with weight 21 Papp, Z , On a continuous one Parameter group of Operator transformations on the field of Mikusihski Operators 229 P a r e i g i s , B . , Non-additive ring and module theory I. General theory of monoids 189 Hbix CMCTCM

t

A

Pareigis,

B . , Non-additive ring and module theory II. ^-Categories, ^-Functors and ^ Morphisms 351 Prasad, B . N . , The Lie derivatives and areal motion in areal space 65 Puystjens, R . , Multiplicative congruences on matrixsemigroups 299 Redei, L.—Märki, L . , Verallgemeinerter Summenbegriff in der/7-adischen Analysis mit Anwendung auf die endlichen p-Gruppen 101 Satyanarayana, M . , A class of maps acting on semigroups 209 Singh, T., Degree of approximation by harmonic means of Fourier—Laguerre expansions . . . 53 Spikes, P. W . — G r a e f , J. R . , Asymptotic properties of Solutions of a second order nonlinear differential equation 39 Szabö, J., Eine Methode zur L ö s u n g von metrischen Aufgaben der Perspektive (Zentralaxonometrie) 97 Szalay, / . , On generalized absolute Cesaro summability factors 343 Capcußoea F. Xp.—Baume ff. ff., 0 6 O A H O M BapwaHTe MeTOAa ycpeAHeHMJi AJIA HennueMHbix CHCTeM HHTerpo AH$(j)epeHUHajibHbix ypaBHCHHM CTaHAapTHoro BHAa 205

Szekelyhidi, L.—Daröczy, Z.—Lajkö, K., Functional equations on ordered fields Upadhyay, M . D . — G u p t a , V. C., Integrability conditions of a structure/ satisfying f ~ X f = 0 Withalm, C., Ober das Ähnlichkeitsprinzip für hyperpseudoholomorphe Funktionen Wunderlich, W., Algebraische Beispiele ebener und räumlicher Zindler-Kurven Bibliographie 3

2

A

BIBLIOGRAPHIE

M . EICHLER, Quadratische Formen und orthogonale Gruppen. 2. Auflage. — H . H . SCHAEFER, Banach Lattices and Positive Operators. — Lie groups and their representations. Edited by I. M . G E L F A N D . — J . D I E U D O N N E , Grundzüge der modernen Analysis, II. — G A B R I E L K L A M B A U E R , Mathematical analysis. — R U D O L F L I N D L , Algebra für Naturwissenschaftler und Ingenieure. — W u Y i HSIANG, Cohomology theory of topological transformation groups. — Numerische Behandlung von Differentialgleichungen. Herausgegeben von R . A N S O R G E , I.

COLLATZ,

G . HÄMMERLIN, W. T Ö R N I G — N A R A Y A N

C . GIRI,

Introduction

to Probability and Statistics Part II: Statistics. — R . V O N R A N D O W , Introduction to the Theory of Matroids. — Die Werke von Jakob Bernoulli. Band 3. Bearbeitet von B. I. V A N DER W A E R D E N , K . K O H L I und J . H E N N Y . — Beiträge zur Numerischen Mathematik 3. Herausgegeben von FRIEDER K U H N E R T und J O C H E N

W. SCHMIDT. — A . M . OLEVSKIJ, Fourier Series with Respect to General Orthogonal Systems. — Husemoller, D . FIBRE

Bundles. — J O N A T H A N

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Localization of noncommutative rings. — M . B R A U N , Differential Equations and Their Applications. — E . H A R Z H E I M — H . R A T S C H E K , Einführung in die allgemeine Topologie. — J . L . K E L L E Y A N D I. N A M I O K A , Linear Topological Spaces. — A . W E I L , Elliptic Functions according to Eisenstein and Kronecker. — J . W E R M E R , Banach Algebras and Several Complex Variables. — C . J . M o z z o c m — M . S. G A G R A T — S . A . N A I M P A L L Y , Symmetrie Generalized Topological Structures. C . J. M O Z Z O C H I , Foundations of Analysis. Landau revisited. — G . P Ö L Y A — G . S Z E G Ö , Problems and Theorems in Analysis. Volume II. — H . J. K O W A L S K Y , Vektoranalysis II. — B. A N G E R — H . B A U E R , Mehrdimensionale Integration. — Beiträge zur Analysis 8. Herausgegeben von R . K L Ö T Z L E R , W . T U T S C H K E , K . WIENER. — NOLTEMEIER, H . , Graphentheorie mit Algorithmen und Anwendungen. — G . N Ö B E L I N G , Einführung in die nichteuklidischen Geometrien der Ebene. — W E R N E R G Ä H L E R , Grundstrukturen der Analysis, I.

77-3523 — Szegedi Nyomda — F . v.: D o b ö Jözsef

173 249 221 289 377

Non-additive ring and module theory I. General theory of monoids By Bodo Pareigis ( M ü n c h e n )

M u c h of the present theory of rings and modules depends only on the multiplicative structure, not on the additive structure of the objects in question. Another part of this theory depends mostly on the additive structure. This last part has extensively been studied in the context of abelian (or additive) categories. We want to introduce categorical tools to study just the multiplicative properties. A surprisingly large part of ring and module theory can be recovered in this way, e.g. Morita theorems, Brauer groups of Azumaya algebras, Maschke's theorem about separable group rings etc. Furthermore this generalized theory applies to a series of examples to be discussed later. The general background of our theory is the notion of a monoidal category (not necessarily Symmetrie or closed or complete or cocomplete). This is a category in which we can form associative "tensor products" of objects and morphisms, with "unit" such that I®X^X^X®L Rings are generalized to monoids in i.e. objects A£%> together with an associative unitary multiplication JX\ A ^ A ^ A . Modules are generalized to ^-objects, i.e. objects together with an associative unitary multiplication v: A & M - + M . We use a Yoneda Lemma like technique to do most computations elementwise. So the associativity of the monoid A , \ i \ A & A - + A can be expressed by a(bc) = (ab)c for "elements" a , b, c of A (see § 1). Similarly an ,4-morphism / : M - + N of ^-objects Af, N is described by f ( a m ) = af(m). This coneept may be applied to a series of examples some of which are rings and modules, coalgebras and comodules, Banach algebras and Banach modules, //-module algebras and modules over them for H a Hopf algebra, monoids (in sets) and monoid sets with equivariant maps etc. In this paper we want to introduce the general background of non-additive ring and module theory, tensor products over arbitrary base-monoids (read: base-rings), and the technique of elementwise computation.

190

B. Pareigis

1. Notation and the universal property of the tensor product Let ,/) be a monoidal ® : # X # — # and an object a:

category, i.e. a category # with and with natural isomorphisms

,4®(£®C) ^ 04®5)®C

/: Q:

I®A^A A(S>I

= A

such that all diagrams >4®(£®(C®Z))) — (/*®5)®(C®Z>) —

((^®^)®C)®Z)

|/4®oc

ja®£>

A