PUBLICATIONES MATHEMATICAE

25. KÖTET

DEBRECEN 1978

ALAPITOTTÄK:

R^NYI ALFRED, SZELE TIBOR £s VARGA O T T O

D A R Ö C Z Y Z O L T Ä N , GYIRES BELA, R A P C S Ä K A N D R Ä S , T A M Ä S S Y LAJOS

K Ö Z R E M Ü K Ö D ^ S E V E L SZERKESZTI:

BARNA BELA

A DEBRECENI TU DOMÄN YEG YETEM M ATEM ATIKAI I NTIzZET E

INDEX

EauHoe, ff.—KoHcmanmuHoe, M . M . , KpaeBbie 3aAami c AByMfl VCJIOBHHMH A-na AH(fc-epeHUMaJibHO—4>yHKu,HOHajibHbix ypaBHeHHö nepßoro nopanKa 187

Das, R . N . — S i n g h , P . , The radius of starlikeness of certain subclasses of analytic functions . . . . 97 Fawzy, L . — M a k a r , R . H . , Oneffectiveness of basicsetsof polynomials associated with functions

ofalgebraic infinite matrices 73 Fenyö, L , Über eine Darstellung linearer Operatoren im Hilbertraum 123 F i e l d l i o u s e , D . J., Semihereditary polynomial rings 211 F i s h e r , B . , On the convolution of distributions 69 F i s h e r , B . , A fixed point theorem for compact metric Spaces 193 G l u s k i n , L . M . — S t e i n f e l d , O., Rings (semigroups) containing minimal (O-minimal) right and

left ideals 275 Grzqslewicz, A . — P o w q z k a , Z . — T a b o r , J., On Cauchy's nucleus 47 G u p t a , V. C.—Upadhyay, M . D . , Some problems on horizontal and complete lifts of <9 (4, — 2)-

structure 77 Györy, K., On polynomials with integer coefficients and given discriminant, IV 155 Györy, K . — P a p p , Z . Z . , Eßective estimates for the integer Solution of norm form and discri­

minant form equations 311 H a f t e n d o r n , D . , Additiv kommutative und idempotente Halbringe mit Faktorbedingung, 1 . . . 107 Istrd(escu, L , Fixed point theorems for some classes of contraction mappings on nonarchimedean

probabilistic metric Spaces 29 Johnson, L . S.—Porter, A . D . — Varineau, V. J., Commutators over finite fields 259 K a l m a r , I. G., Atomistic orthomodular lattices and a generalized probability theory 139 Kätai, L , On additive functions 251 Kenny, G. O., The archimedean kernel of a lattice-ordered group 53 Koncmaumimoe, M . M . — B a i m o e , ff. JJ., KpaeBbie 3aAa4H c AßyMH VCHOBHAMH RJW AH<{)-

4>epeHUMajibH0—4)yHKUHOHanbHbix ypaBHeHHö nepßoro nop AKa 187 Kormos, J . — T e r d i k , Gy., Notes on matrix-valued stationary stochastic processes 1 281 K u m a r , V., Some extended Hermite—Fejer interpolation processes and their convergence . . . . 285 Lajkö, K., Generalized diüerence-property for functions of several variables 169 Lajos, S.—Szäsz, F . , On (w, //)-ideals in associative rings 265 Lajos, S.—Szäsz, G., On characierizations of certain classes of semigroups 275 M a k a r , R . H — F a w z y , L . , On effectiveness of basic sets of polynomials associated with functi­

ons of algebraic infinite matrices 73 Matusü J., Der rechts- und linksseitige Wert einer Distribution im Punkte 89 M a u r e r , I. Gy.— Viräg, /., Sur les classes de tolerance d'une algebre universelle 237 M i s h r a , R . S . — S i n g h , R . N . , On birecurrent Kahler manifold 61 Müller—Pfeiffer, E . , Eine Abschätzung für das lnfimum des Spektrums des Hillschen Differenti-

aloperators 35 N e m e t h , J., Note on imbedding theorems 291 Pande, H . D . — S h u k l a , H . S., Projective symmetry in a special Kawaguchi Space 117 Papp, Z.—Györy, K., Eöective estimates for the integer Solution of norm form and discriminant

form equations 311 Pareigis, B . , Non-additive Ring and Module Theory III. Morita equivalences 177 Porter, A . Z>.— Varineau, V. J.—Johnson, L . S., Commutators over finite fields 259 Powqzka, Z . — T a b o r , J.—Grzqslewicz, A . , On Cauchy's nucleus 47 Schierwagen, A . , Bemerkungen zu einem Teilerproblem 41 S h u k l a , H . S.—Pande, H . D . , Projective symmetry in a special Kawaguchi space 117 Shultz, H . S., A new albegra of distributions on R n 85 Siddiqi, J a m i l A . , The determination of the jump of a function by Nörlund means 5

Sigmon, K., A universal construction of universal coeflicient sequences 21 Singh, P.—Das, R . N . , The radius of starlikeness of certain subclasses of analytic functions . . . 97 Singh, R . , A sufficient condition for univalence 1 Singh, R . N . — M i s h r a , R . S., On birecurrent Kahler manifold 61 Steinfeid, O.—Gluskin, L . M . , Rings (semigroups) containing minimal (0-minimal) right and

left ideals 275 Stern, M . , Standard ideals in matroid lattices 241 Szäsz, F.—Lajos, S., ( m , n)-ideals in associative rings 265 Szäsz, G.—Lajos, S., On characterizations of certain classes of semigroups 275 Szäz, Ä., Periodic generalized functions 85 Szolcsänyi, E . , Hyperflächen mit Riemannscher Maßbestimmung im Finslerschen Raum 11 . . . 203 Tabor, J.—Grzqslewicz, A.—Powqzka, Z . , On Cauchy's nucleus 47 Terdik, Gy.—Kormos J., Notes on matrix-valued stationary stochastic processes 1 281 Upadhyay, M . D . — G u p t a , V. C, Some problems on horizontal and complete lifts of 0 (4, - 2)-

structure 77 Varga, T., Über Berwaldsche Räume 1 213 V a r i n e a u , V.J.—Johnson, L . S.—Porter, A . D . , Commutators over finite fields 259 Vincze, E . , Über die Einführung der komplexen Zahlen mittels Funktionalgleichungen 303 Viräg, I . — M a u r e r , I . Gy., Sur les classes de tolerance d'une algebre universelle 237 Widiger., A . , Die Ringe, deren sämtliche Unterringe Hauptrechtsidealringe sind 195 Wiegandt, R . , Correction and remark to my paper "Characterizations of the Baer radical class

by almost nilpotent rings" 327

BIBLIOGRAPHIE

TOM M . APOSTOL, Introduction to Analytic Number Theory. — TOM M. APOSTOL, Modular Functions and Dirichlet Series in Number Theory. — L. CESARI—R. KANNAN—J . D. SCHUÜR, Editors, Nonlinear Functional Analysis and Differential Equations. — J . BERG—J. LÖFSTRÖM, Interpolation Spaces. — N. J . PULLMAN, Matrix Theory and its Applications: Selected Topics. — Functional Analysis: Proceedings of the Bra-zilian Mathematical Society Symposium. Edited by DJAIRÖ GUEDES DE FIGUEIREDO. —JUAN JORGE SCHÄFFER, Geometry of Spheres in Normed Spaces. — J . DONALD MONK, Mathematical Logic. — R. GABASOV—F. KIRILLOVA, The Qualitative Theory of Optimal Processes. — SERGE LANG, Introduction to Modular Forms. — M. K . AGOSTON, Algebraic Topology: a firstcourse. — R. E. CHANDLER, Hausdorff Compactifications, — W . V . VASCONCELOS, The rings of dimension two. — ST. P. FRANKLIN—B. V . SMITH THOMAS, Topology. — LARRY E. MANSFIELD, Linear Al­gebra with Geometrie Applications. — P. LAX—S. BURSTEIN—A. L A X , Calculus with Applications and Computing. I. — G. B. SELIGMAN, Rational Methods in Lie Algebras. — D. G. NORTHCOTT, Finite Free Resolutions. — KIRILLOV, A. A., Ele­ments of the Theory of Representations. — R. C. GUNNING, Riemann Surfaces and Generalized Theta Functions. — J . G. KEMENY—J . L. SNELL, Finite Markov Chains. — R. ENGELKING, General Topology. — J . G. KEMENY—J . L. SNELL—A. W . KNAPP, Denumerable Markov Chains. — HAROLD R. JACOBS, Geometry. — 3. OpHfl—M. ITacTop—M. IleftMaH—IL PeBec—M. Pyaca, Majiaa MaTe-MaTHHecKaa 3HUKKJioneaH5i 329—341

78-5087 — Szegedi Nyomda — F. v.: D o b ö Jözsef igazgatö

Non-additive Ring and Module Theory, III. Morita Equivalences

Dedicated to the memory of Andor Kertesz

By B. PAREIGIS (München)

This paper is a continuation of [19], [20]. References are quoted there. As in ring theory one may ask the question when two categories A<& and B #

for monoids A and B are equivalent. Now in ring theory we know from the additivity of the equivalences and that the natural bijections

H o m ß ( ^ ( M ) , N ) 2* U o m A ( M , 9 ( N ) ) and U o m A { ^ ( N \ M ) ~ Hom ß(AT, & ( M ) )

are isomorphisms of abelian groups. So in the general case we only want to con-sider equivalences such that there are isomorphisms B [ ^ ( M ) , N ] ^ A [ M 9 &(N)] and A [ & ( N ) 9 M ] ^ B [ N 9 J*(M)]. In view of theorem 4.3. this is equivalent to study-ing equivalences such that and 0 are ^f-functors. The last condition can be studied even in monoidal, non-closed categories.

We call $ and B # ^ - e q u i v a l e n t i f there are inverse equivalences fF'.jfß^ßl and <S:B<g-+A<g such that & and <& are ^-functors.

Without loss of generality we shall only consider equivalences and ^ to-gether with isomorphisms Q i ^ t f ^ I d and XF: g & ^ I d such that ^ *F = and *F<g = <g<P. Then $ and *F and their inverses are already adjointness morphisms.

5.1. Theorem. L e t <ß be a n a r b i t r a r y m o n o i d a l c a t e g o r y . L e t a n d y'-ßß^A^ o e Averse - e q u i v a l e n c e s . Then t h e r e a r e o b j e c t s P£.A€B

a n d Q€B%A such t h a t

a) t h e r e a r e n a t u r a l i s o m o r p h i s m s

&(M)^Q®AMs,A[P,M] i n A<€9

9 ( N ) ~ P®BN^ B [ Q , M ] i n B%

a n d P B - c o f l a t a n d Q A - c o f l a t . b) t h e r e a r e i s o m o r p h i s m s of A — A - r e s p . B — B - b i o b j e c t s

A =* P®BQ a n d B^Q®AP such t h a t t h e d i a g r a m s

P<S*(Q<S>P) = (P<&Q)®P * A®P \ \

P®B * P

12D

178 B. Pareigis

a n d

Q

B [ Q , B ] ^ P i n A < g B ,

A [ P , A ] ~ Q i n jßAi

d) t h e r e a r e i s o m o r p h i s m s

B [ Q , Q] ^ A i n A%>A a n d as m o n o i d s ,

Ä [ P , P] ^ B i n B^B a n d as m o n o i d s .

PROOF. By the symmetry of the Situation we only have to prove one half of the assertions.

There is an isomorphism F ' ( M ) ^ Q < g > A M natural in M by Theorem 4.2 since fF is a ^-functor and clearly preserves difference cokernels as an equivalence. By the same theorem we have & ( N ) ^ B [ Q , N] natural in N , since ^ is adjoint to !F. This proves a).

We have an isomorphism A^&&r(A)^P®B(Q®AA)^P®BQ in Further-more we have a commutative diagram

hence A^P®BQ as A — ,4-biobjects. The adjunction morphism XF\ ^ t F ^ I d induces the evaluation morphism P®BA[P,M]^M with V{p®f) = { p ) f . By definition of the isomorphism

A^P®ßQ we get a commutative diagram

Hence i f the isomorphism P < g > B Q ^ A is described by p®Bq^pq and the morphism Q ® A M ^ A [ P , M] is given by

Now i f M ^ A ^ B then we get (p)<p (q®Amb) = ( p q ) ( m b ) = ( ( p q ) m ) b — ( ( / ? ) c p (q®Amj)b = r=(p)((P(a®Am)b)> hence Q®AM and A [ P , M] are isomorphic as B—Z?-biobjects.

A®A ^ P®B(Q®A(A®A)) ^ P®B(Q®A)

A ^ P®B(Q®AA) P®BQ

q®Am (p(q®Am), we get ( p q ) m = (p)q>(q®Am).

Non-additive Ring and Module Theory, III. Morita equivalences 179

In particular we get A [ P , A ] ^ Q as B—,4-biobjects and A [ P , P ] ^ Q ® A P ^ B as B—i?-biobjects.

To prove the monoid isomorphisms we first observe that

P®(Q®P) = (P®Q)®P - A®P and I \

P®B - P

Q®(P®Q) ~ (Q®P)®Q - B®Q

Q®A > Q

commute. This follows from ^ ^ Q ® A , < g ^ P ® B and from the fact that # J j r : J z : W - J z r and ^ r ^ ^ - J ^ r e s p . ^ < P \ <8 and U * ^ : ^ ^ - ^ are equal. So we get

<P><K<7WM?W) = <j>q'Wq")p" = P W P ' W P " ) -

Since the isomorphism A | 7 \ is given by

A [ P , P] — Q®AP = B

or cp(q'®p')^-+q''p\ the composition cp(q''®p')(p(q"®p") is mapped to the product ( l ' p ' ) ( f p " ) ' I*> ( ? ' ® / 0 is the identity then p ( q ' p ' ) = p for all p . But ^[P, P ] - + B is an isomorphism hence q ' p ' — X ^ B .

5.2. Corollary: 77ze morphisms

P ( X ) X A [ P , A ] ( Y ) 3 (p , / ) - (p)f£A(X®Y) a n d

A [ P , A ] ( X ) X P ( Y ) 3 (/, p ) ~fp£A[P, P](X®Y)

w i t h ( p ) ( f p ) ' = ( ( p / ) f ) p i n d u c e isomorphisms

P®BA[P, A ] ^ A and A [ P , A]®AP ^ A [ P , P ] ,

The analogous assertions h o l d f o r Q a n d B . PROOF. The first isomorphism, the evaluation morphism, was discussed in

the proof of 5.1. The second isomorphism is just given by

ÄP, A}®AP SSJU Q®AP - J L - A [ P , P ] .

We have seen that each ^-equivalence is induced by some object P ^ A ^ B with the properties of Corollary 5.2. The converse will proved after a more detailed study of the properties exhibited in Corollary 5.2.

A n object P £ A % will be called finite, i f A [ P , A] and B : = A [ P 9 P] exist, if P is 2?-coflat and A [ P , A ] is /4-coflat and i f the morphism A [ P , A ] ® A P - + A [ P , P] induced by A [ P I A } ( X ) X P ( Y m f p ) * f p t A [ P 9 P ] ( X ® Y ) with { p ' ) f p = { { p ' ) f ) p is an isomorphism. P will be called f a i t h f u l l y p r o j e c t i v e i f it is finite and i f the morphism P ® B A [ P , A ] - + A induced by the evaluation is also an isomorphism.

12*

180 B. Pareigis

5.3. Theorem. Let A , B be monoids i n (ß9 P ^ A ^ B B - c o f l a t a n d Q^߀A A - c o f l a t . Given morphisms f : P < g ) B Q - + A i n A%>A a n d g : Q < g > A P - + B i n B^B such t h a t the d i a g r a m s

f®Ap P ® B ( Q ® A P ) = ( P ® B Q ) ® A P — Ä®ÄP

P®BB P

Q ® A ( P ® B Q ) = ( Q ® A P ) ® B Q ^ ' B ® B Q

Q ® A A Q

commute. Assume t h a t there is P O ^ B ^ O ^ P ^ B Q ^ ) s u c n t n a t P o a o : = f ( P o < ^ B ( l o ) =

= l£A(I). Then f is an i s o m o r p h i s m . Assume t h a t i n a d d i t i o n there is qi®APi£Q®AP(I) such t h a t q1p1:=g(q1®ÄpJ = l£B(I). Then P ® B : B < # - + A V a n d Q ® A : Jß-^ß^ a r e inverse -equivalences. I n p a r t i c u l a r P ^ A ^ a n d Q ^ B ^ a r e

f a i t h f u l l y p r o j e c t i v e .

PROOF. Define f ' : A - + P ® B Q by f ' ( a ) = a p 0 ( g ) B q 0 . Then f f ' ( a ) = a p Q q 0 = a and f'f(p®Bq) = ( p q ) p 0 ® B < l o = P(m)<8>B<Jo = />®B(0PO)?O = P®B<](PO<IO) = P®B<1-Hence / is an isomorphism.

Furthermore the functors P ® B Q ® A = A®A and Q ® A P ® B — B ® B are both isomorphic to the identity-functors on /ß resp. B

cß9 hence they are inverse equiv­alences. Furthermore P<g>B and Q ® A are ^-functors by Theorem 4.2.

5.4. Theorem. Let P ^ A ^ be f a i t h f u l l y p r o j e c t i v e . Then A [ P 9 - ] : A V - + A J > t J > { € exists a n d is a ^-equivalence.

PROOF. By definition A [ P , A ] and A [ P , P] = B exist. Furthermore P Z ^ ß , Q:=A[P9A]£B

(£A and the hypotheses of Theorem 5.3. are satisfied by the very definition of Q ® A P - + B and P < g > B Q ^ A . So Q<8>A: j^-^ßß is a ^-equivalence. By Theorem 5.1. we get Q®A^A[P9 - ] .

Let us now apply our theorems to the case where the tensor-product in is the (direct) product (example c) of § 1). Furthermore assume that each canonical epimorphism M X N ^ M X A N induces a surjective map M X N ( I ) - + M X A N ( I ) . This is for example the case i f I i s projective in the category cß. We say that M X N - * -- + M X A N is r a t i o n a l l y s u r j e c t i v e . Assume that A<ß and ß ^ are ^-equivalent by P ^ ß — ' - B ^ ^ A ^ a n c * Q ® A ' A ^ ^ B ^ ' Then we have surjective maps f : P ( I ) X X Q ( I ) ^ P X Q ( I ) ^ A ( I ) and g : Q ( I ) X P ( I ) ^ Q X P ( I ) ^ B ( I ) suchthat ( p q ) p ' = = P ( q p ' ) and ( q p ) q / = q ( p q / ) i f f ( p , q ) = p q and g ( q , p ) = q p . Let p £ P ( I \ <7i€ß(/), *=0, 1 be chosen such that p 0 q 0 = l £A(I)9 1

Let us now assume that each element in A (I) which has a left inverse has also a right inverse. We wish to show A ^ B as monoids. First we show p1q1=l£A. By definition we have ( p 0 q±) { p x q0) = p 0 fapjqo =p0q0=l£A (/), hence ( p x q0) ( p 0 qx) = l. Furthermore we have P o q x P ^ P o - This implies /?i#i=l * P i q i = ( P i q o P o q i ) P i q i = = P i ^ ( P Q a i P i ) a i = P i a Q P o a i = ^ ^ A i J ) ' Now define morphisms P { X ) ^ p ^ p q x ^ A ( X ) and A { X ) ^ a ^ a p X ^ P . They are obviously mutually inverse morphisms in A

<€. Hence B ~ A [ P , P ] ~ A [ A 9 A ] ~ A .

As a special case we get

Non-additive Ring and Module Theory, III. Morita equivalences 181

5.5. Corollary: I n t h e c a t e g o r y of sets £f w i t h t h e p r o d u c t as m o n o i d a l c a t e g o r y l e t A be a g r o u p or a c o m m u t a t i v e m o n o i d o r finite. T h e n A S f a n d B S f a r e e q u i v a l e n t iff A ~ B .

PROOF. In Sf the morphism-sets form an inner hom-functor, so by Theorem 4.3. each equivalence A£f^B£f is an ^-equivalence. Furthermore {0} is projective in Sf. If A is a group or commutative or finite then each element which has a left inverse in A (I) has also a right inverse. So all conditions of the previous discussion are satisfied. Hence A ^ B . The converse is trivial.

The central part of the Morita Theorems

For this section we will always assume that # is a Symmetrie monoidal category. Let us consider ^-funetors " K i ^ ^ j f g such that there are P , Q ^ B ^ A w i t h ^-isomorphisms $/^P®A, ir^Q®A.

Define a new ^-funetor °U®Y for Y^fß by <%®Y(M):=W(M®Y)^ ^ P ® A ( M ® Y ) . Because of the symmetry of V we have Ol® Y ( M ) ^ ( P ® Y)®AM hence $1 ®Y indeed is a ^-funetor.

Define [< , iT](F):=#-Mor (°H® Y, V ) as the set of ^-morphisms from <%®Y t o r. For h : Z - + Y in <g define [ % TT](A): [ « , r T J i ( Y ) ^ [ % r \ ( Z ) by [<%,r](h)((p)(M):=(W(M®Z) * ( M » * ) . Then [ « , 1T] is a contravariant funetor from # to the category of sets.

6.1. Theorem. T h e r e is a n a t u r a l i s o m o r p h i s m of f u n e t o r s f r o m <ß t o t h e c a t e g o r y of sets:

[ % r)(Y)s<BVAP®Y,Q).

I f B [ P , Q ] A exists then [*, r ] ( Y ) ^ B [ P , Q ] A ( Y ) .

PROOF. Let f€B<gA(P® Y, Q ) . Then define r ~ \ ( Y ) by

cp(M) := (W(M®Y) s ( P ® Y ) ® A M F % A M • Ö ^ A f as 1^(A/)).

For g t / ^ i M , N ) we get a commutative diagram

%(M®Y) ^ (P(g)y)(8)AM — - — - Ö<8uM ^ y ( M )

I (P®Y)®Ag Q®Ag n g )

W(N®Y) = ( P ® y ) ® ^ ^ / < 8 > i i " . Q®AN = ^ N )

hence <p is a natural transformation from °U®Y to Furthermore the following diagram commutes:

U m X ) * Y ) Z ( P * Y ) 9 A ( M * X ) Q<»A(M<»X)Zf(M<,X)

\\\ Iii III III

WMQYteXZ((Pe>Y)®AM)®X ( ^ i X ( Q s i M ) * X ^ j / ( M ) 0 X

182 B. Pareigis

Hence <p is a ^-morphism. This defines a map

X:BVA(P®Y,Q)-[%r](Y).

Conversely let (p£[<%, r~](Y) and define f^B^A(P®Y, Q) by / : = {P®Y^P®AA®Y^{A®Y)-^^-r(Ä)^Q®AA^Q). Clearly f^B^(P<g>Y, Q). To show that fdB^A{P®Y,Q) consider the following commutative diagram

?*A(A<9Y)®A *((P*Y)0aA)QA2U(A*Y)®A ^(A)<8ty(A)<»Ag(Q*AA)«A

n wi \w m » m P*Y)»A =(P»Y)»A(A<»A)£ U((AeA)®Y) ^

(A^y(A»A)za»A(A*A)*Q«A

U ( / i A * Y ) U f r

?<*Y & (P®r)0AA = U(A<9Y). yCA) £ Q®AA £ Q

where the morphism from ( P ® Y)®A to Q®A along the upper side of the diagram is just f®A and the morphism from P ® F to Q along the lower side is / . Hence / is a right /1-morphism. So we have a map

Now

and

n:[% r](Y)^BVA(P®Y9Q).

n i ( f ) = (P®Y^= 1?{A®Y)

( P ® 7 - (P®Y)®AA f®AA

r ( A ) = Q) =

> Q®AA =Q)=f

z n ( ( p ) ( M ) = ( # ( A / ® r ) s ( p ® y ) ® , 4 M * ( < P ) ^ M _ . Ö ^ . M ~ T^(A/)) =

= ( « ( A f ® r ) ^ ( / 4 ® 7 ) ® I 4 M r(A)®AM ~ - T ( M ) ) = cp(M)

since is a ^-morphism. Hence we have [%iir](Y)^B

cßA(P®Yi Q ) . It remains to show that this isomorphism is a natural transformation. Let

h \ Z - + Y be in <g. Then

B<$A(P®Z, Q) HZ)

1% r](X)

*1(Z)

Non-additive Ring and Module Theory, III. Morita equivalences

commutes since for ' A ( P ( g ) Y , Q) we have

183

U(M<S>Z) £ ( P $ Z ) < 8 > A M

U ( M * > k )

commutative and thus

([•, r ] ( h ) o Z ( Y ) ) ( f ) ( M ) =

= («(A/®Z) ^ ( M 0 / I ) - ®(M®Y) ~ (P®Y)®AM £2±?L+ Q®AM =* r ( M ) ) =

= (<#(M®Z) s ( P ® Z ) ® ^ M - — * ß ® u M s f(M)) =

= ( I (Z)o B ^(P®Ä, 0)(/)(M). For this theorem we have two applications. Before we discuss them, we have

to introduce the notion of the center of a monoid. It is clear that A

cß(A9 A ) ^ A ( I ) as monoids in the category of sets. The iso­morphism is given by

A V ( A , A ) 5 f ~ f ( l ) = M A ( I )

A ( I ) ^ a — ( A ( X ) 3 b — ba£A(X))eA<g(A9 A ) .

Now those elements a£A(I) which commute with all b ^ A ( X ) for all X induce in A

C€(A9 A ) precisely the A — -morphisms AßA(A9 Ä)9 which then is a commutative monoid. So a possible definition of the center of A could be A

C 6 A \ A 9 Ä). But this is only a set, not an object in (€. A possible generalization to an object in # is A [ A 9 A ] A if this exists. If it does not exist we know at least the funetor represented by this object. So we define the c e n t e r of A as a funetor from # to Sf\ the category of sets, by Cent (A)(X):=A<gA(A®X9 A ) . If A [ A 9 A ] A exists we have Cent ( A ) ( X ) s z = A [ A 9 A ] A ( X ) .

As in § 3 Cent (A) can only be defined in a Symmetrie monoidal category in contrast to A V A ( A 9 Ä). In § 2 we showed that and c€^cel hence # ( / , / )~ ^{€1(J9 I ) is a commutative monoid [18, Theorem 1] for (possibly nonsymmetric) monoidal categories.

Let A be a monoid in a Symmetrie monoidal category Let A ! d : A

X 8 - + A

( # denote the identity funetor. Then A I d is clearly a ^-funetor and AId^A®A as ^-funetors.

184 B. Pareigis

6.2. Theorem. Cent ( A ) ^ [ J d , A I d \ .

P R O O F . Cent ( A ) ( X ) = A < $ A ( A ® X 9 A ) ^ [ A I d 9 A I d ] ( X ) . The isomorphism of Theorem 6.2. is only an isomorphism of objects in (€.

But there is an additional structure, a multiplication on these functors. If they were representable the representing objects in # would be monoids. The multi-plicative structure on A [ A 9 A ] A as it has been studied in § 3 is reflected in A<ßA(A®X9 A ) by the commutative diagram

A [ A 9 A ] A ( X ) X A [ A 9 A ] A ( Y ) - A [ A 9 A]A(X®Y) II* II?

Ä (A ® X 9 A ) X Ä ( A ® Y 9 A ) - + AVA (A ® X® Y9 Ä)

where the lower map is given by

( / , g ) - ( ^ W ^ A ® Y A ) .

The unit is described by

A®I^ A ) £ A < # A ( A ® X 9 A ) .

[ A I d 9 A I d ] carries a multiplaciticve structure via

[ A I d , A I d ] ( X ) X [ A I d , A I d ] ( Y ) [ A I d , AId]{X®Y)

by T(<p, ip) = (AId®X®Y *®r • A I d ® Y A I d )

and there is a unit

I ( X ) 5 f ~ ( A I d ® X - ^ * AId®I~ AId)e[AId9 A I d ] ( X ) .

Using the isomorphism of Theorem 6.1. it is easy to see that they are compatible with the multiplication and the unit map. Hence the isomorphism of Theorem 6.2. is a „monoid isomorphism".

6.3. Corollary: Let A<ß a n d jß be <€-equivalent. Then Cent (4) Cent ( B ) as f u n c t o r s f r o m to £f. If both f u n c t o r s a r e representable then the W o r e p r e s e n t i n g objects a r e i s o m o r p h i c as commutative monoids i n

A [ A 9 A ] A - B [ B 9 B ] B .

P R O O F . We show [ A I d 9 A I d ] ^ [ B I d 9 B I d ] . Let & \ ß # the given ^-equivalence. First we show

[ A I d 9 A I d ] ( X ) ~ [ ^ 9 & ] ( Y ) 9

or #-Mor (AId® Y9 A I d ) 9 £ V - M o T { P ® Y9 9 ) .

Let (p : A I d ® Y - + A I d be a natural transformation. Define ^ocp: <F® Y - + 2 F by ^ o c p ( M ) : &{M® 7 ) - ^ ( A f ) as & r o c p ( M ) = ^ ( c p ( M ) ) . Since & is an equivalence it is clear that cp^^ocp is a bijection between the sets of natural transformations. Now we show that cp is a ^-morphism iff J ocp is a ^-morphism. cp is a ^-morphism iff the diagrams

Non-additive Ring and Module Theory, III. Morita equivalences 185

III IG M®Y®X VW®* » M®X

commute. #o<p is a ^-morphism iff the outer diagrams of

II? Iii

commute where the lower part commutes in any case since & is a ^-funetor. But the upper part commutes iff the previous diagram commutes. Hence

% - M o r ( A I d < S ) Y , A I d ) ^ < g - M o v ( 3 ?

Now we show [ B I d , B I d ] ( Y ) ^ [ ^ , & ] ( Y ) or

* - M o r ( B W ® y , B / d ) ~ V - M o r ^ ® } ^ ) .

It is clear that the correspondence between cp: BId®Y-+ B I d and c p ^ o \ &®Y^fF with

( ( p o ^ ) ( M ) : = (&(M®Y) s &(M)®Y • #"(A/))

induces an isomorphism between the sets of natural transformation, since SF is an equivalence. Furthermore cp is a ^-morphism iff the diagrams

in ii

186 B. Pareigis: Non-additive Ring and Module Theory, III. Morita equivalences

commute. On the other hand c p o ^ is a ^-morphism iff the outer diagrams

III

commute. The first and third part commute by definition. In the middle part take into account that SF is a ^-funetor. Then it commutes iff the previous diagram for <p commutes. Hence [ B I d , B I d ] ( Y ) ^ , 3 F ] ( Y ) ^ [ A I d 9 J d ] ( Y ) .

The reader can easily verify that these isomorphisms are natural isomorphisms in Y. Furthermore they preserve the multiplication" given by composition of morphisms just before Corollary 6.3. They also preserve the „unit" . Hence

A [ A , A ] Ä - B [ B , B ] B

as monoids (if they exist) or

Cent 04) ^ Cent(P)

with the multiplicative structure. 6.4. Corollary: L e t be the u n d e r l y i n g f u n e t o r . Then [ < % 9 < % ] ( Y ) ^

^ A o p ( Y ) n a t u r a l i n Y^ß a n d c o m p a t i b l e w i t h the m u l t i p l i c a t i o n on b o t h sides.

PROOF. By Theorem 6.1. and the fact <%?*A®Ä we have <&](Y)2z ^A{A® Y, A ) ^ A o p ( Y ) as left multiplications and these isomorphisms are natural in Y and compatible with the multiplication.

So we have seen that just from the knowledge of the underlying funetor % we may regain the monoid A up to an isomorphism.

(Received November 2 0 , 1 9 7 5 . )

Bibliography

[19] B. Pareigis, Non-additive Ring and"module Theory I. General Theory of Monoids. P u h l . M a t h . (Debrecen) 24 (1977), 189—204.

[20] B. Pareigis, Non-additive Ring and Module Theory II. ^-Categories, ^-Functors and ^-Morphisms. Publ. M a t h . (Debrecen) 24 (1977), 351—361.