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PUBUCATIONES MATHEM ATICAE
2 4 . K Ö T E T
D E B R E C E N
1977
A L A P I T O T T Ä K :
RE-NYI ALFRED, SZELE TIBOR Es VARGA O T T O
D A R Ö C Z Y Z O L T Ä N , GYIRES BEXA, R A P C S Ä K A N D R Ä S , TAMÄSSY LAJOS
K Ö Z R E M Ü K Ö D E S ^ V E L S Z E R K E S Z T I :
BARNA BELA
A DEBRECENI TUDOMÄNYEGYETEM MATEMATIKAI INTEZETE
INDEX
Baünoe JJ. JJ.—Capa(fioea r . Xp., 06 OAHOM BapnanTe MeTO a ycpeflHeHHH RJW HejiHHeii-HWX CHCTCM HHTerpo flH^epemrHajibHferx ypaBHeHHÄ CTaH/japTHoro BH^a 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 GUmer, R . , Modules that are finite sums of simple submodules 5 Glevitzky, B . , On polynomial regression of polynomial and linear statistics 151 Graef, J. R.—Spikes, P . W., Asymptotic properties of Solutions of a second order nonlinear dif-
ferential equation 39 Gregor, J., Tridiagonal matrices and functions analytic in two half-planes 11 G u p t a , V. C.—Upadhyay, M . D . t Integrability conditions of a structure f A statisfying
p - p 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 J a k d l , L . , Über 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, /., 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 signa-
tures 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.—Mich/er, G., On products of füll linear rings 123 Leavitt, W. G.—Enersen, 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 , G y . , On the functional equation f ( x + y ) + g ( x y ) = h ( x ) + h ( y ) 25 M d r k i , L . — R e d e i , L . , Verallgemeinerter Summenbegriff in der /?-adischen Analysis mit Anwen
dung auf die endlichen /?-Gruppen 101 M i c h l e r , G.—Lambek, J., On products of füll linear rings 123 M u 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 Nguyen Xuan 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
Pareigis, B . , Non-additive ring and module theory II. ^-Categories, #-Functors and <€-Morphisms 351
Prasad, B . N . , The Lie derivatives and areal motiorr in areal space 65 Puystjens, R . , Multiplicative congruences on matrixsemigroups 299 Redei, L.—Märki, L . , Verallgemeinerter Summenbegriff in der /?-adischen Analysis mit Anwen
dung auf die endlichen /?-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.—Graef, J. R . , Asymptotic properties of Solutions of a second order nonlinear dif-
ferential equation 39 Szabö, J„ Eine Methode zur Lösung von metrischen Aufgaben der Perspektive (Zentralaxono
metrie) 97 Szalay, L , On generalized absolute Cesaro summability factors 343 Capacßoea r. Xp.—Baume ff. ff., 06 OAHOM BapwaHTe MeTOfla ycpe HeHHH AJIA HenHHeii-
Hbix CMCTeM HHTerpo AH(J) epeHUHajibHbix ypaBHCHHö CTaHAapraoro Bw a 205 Szekelyhidi, L.—Daröczy, Z.—Lajkö, K., Functional equations on ordered fields 173 Upadhyay, M . D . — G u p t a , V. C, Integrability conditions of a structure/A satisfying p - X 2 f = 0 249 W i t h a l m , C, Ober das Ähnlichkeitsprinzip für hyperpseudoholomorphe Funktionen 221 Wunderlich, W., Algebraische Beispiele ebener und räumlicher Zindler-Kurven 289 Bibliographie 377
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 1. M. GELFAND.—J. DIEUDONNE, Grundzüge der modernen Analysis, II. — GABRIEL KLAMBAUER, Mathematical analysis. — RUDOLF LINDL, Algebra für Naturwissenschaftler und Ingenieure. — Wu Yi HSIANG, Cohomology theory of topological transformation groups. — Numerische Behandlung von Differentialgleichungen. Herausgegeben von R. ANSORGE, I. COLLATZ, G. HÄMMERLIN, W. TÖRNIG—NARAYAN C. GIRI, Introduction to Probability and Statistics Part II: Statistics. — R. VON RANDOW, Introduction to the Theory of Matroids. — Die Werke von Jakob Bernoulli. Band 3. Bearbeitet von B. I. VAN DER WAERDEN, K. KOHLI und J. HENNY. — Beiträge zur Numerischen Mathematik 3. Herausgegeben von FRIEDER KUHNERT und JOCHEN W. SCHMIDT. — A. M. OLEVSKJJ, Fourier Series with Respect to General Orthogonal Systems. — Husemoller, D. FIBRE Bundles. — JONATHAN S. GOLAN, Localization of noncommutative rings. — M. BRAUN, Differential Equations and Their Applications. — E. HARZHEIM — H. RATSCHEK, Einführung in die allgemeine Topologie. — J. L. KELLEY AND I. NAMIOKA, Linear Topological Spaces. — A. WEIL, Elliptic Functions according to Eisenstein and Kronecker. — J . WERMER, Banach Algebras and Several Complex Variables. — C. J. MOZZOCHI—M. S. GAG RAT—S. A. NAIMPALLY, Symmetrie Generalized Topological Structures. C. J. MOZZOCHI, Foundations of Analysis. Landau revisited. — G. PÖLYA— G. SZEGÖ, Problems and Theorems in Analysis. Volume II. — H. J. KOWALSKY, Vektoranalysis II. — B. ANGER—H. BAUER, Mehrdimensionale Integration. — Beiträge zur Analysis 8. Herausgegeben von R. KLÖTZLER, W. TUTSCHKE, K. WIENER. — NOLTEMEIER, H., Graphentheorie mit Algorithmen und Anwendungen. — G. NÖBELING, Einführung in die nichteuklidischen Geometrien der Ebene. — WERNER GÄHLER, Grundstrukturen der Analysis, I.
77-3523 — Szegedi Nyomda — F. v.: D o b ö Jözsef
Non-additive ring and module theory II. # - Categories, # - Functors and * - Morphisms
By B. PAREIGIS (München)
Consider the example (Cat, X , 1) of a monoidal category. A monoid in this category is a (small) category ^ together with functors ® r ^ X ^ 7 - ^ and Z T \ 1—# (where we use the notation .5 r(e)=/) such that the diagrams
commute. This means A®(B®C) = (A®B)®C for all A , B , C ^ and A ® I = = A = 1 ® A for all A^ß. These identities are natural transformations in A , B , C . Such a category is called a st r i e t l y m o n o i d a l category. For the general case the two diagrams (*) are commutative up to a natural isomorphism and such that the coherence conditions of § 1. hold. By thecoherence theorem of [5] this implies that all morphisms composed of a : A < g > ( B < & C ) ^ ( A < g > B ) < g > C , X : I ® A ^ A , Q : A ® I ^ A , identities and ® which formally have common domain and codomain are equal.
Now consider an object Jt in (Cat, X , 1) on which a strict monoidal category ^ operates in the right. Then J i is a category together with a funetor ® '.MYs^^M^ such that
These identities are natural transformations in M , C , D . Such a category J i will be called a s t r i c t ^-category. A useful generalization of this is a %>-category J i for an arbitrary monoidal category c€. This is a category J i together with a funetor ^ ' . J i X ^ - ^ J i and natural isomorphisms ß \ M ® ( C ® D ) ^ ( M ® C ) ® D and
o \ M ® I ^ z M such that all formal diagrams composed of a,ß,a,X,Q9 their in-verses, ® in and ® with respect to J i > identities, and compositions commute.
M ® ( C ® Z ) ) = ( M ® C)®£> for all M £ J ( 9 C , D £ %
and
M ® / = M for all M ^ J i .
352 B. Pareigis
In particular we require the commutativity of the following diagrams:
M ® ( C ® ( Z ) ® £ ) ) • * ( M ® C ) ® ( Z > ® £ ) j M ® a \ ß
M ® ( ( C ® £ ) ® £ ) JL (M®(C®D))®E^((M®'C)®D)®E
M® (I ® C ) -—— (M® I ) ® C
M<sA\ / ^ C
M ® C
If is a Symmetrie monoidal category (which corresponds to the notion of a com-mutative monoid in Cat), then we also require the commutativity of
N o w let us regard the corresponding morphisms. A morphism of monoids in Cat is a funetor S F ^ — O ) of strictly monoidal categories # and such that
and ^(C®D) = &{C)®&(D)
Non-additive ring and module theory IL 353
where I^ß and J ^ Q ) are the neutral objects. The identities are natural transformations in C and D . If # and 3 are just monoidal categories then we require natural isomorphisms
b\3F(C®D) s* y(C)®P(D)
C : ^ ( / ) s 7
such that the following diagrams commute:
p cc® i) fco» r c n
> ( r ® c)
; F ( C ® ( Z ) ® £ ) ) - i ^ ( C ) ® ^ ( Z ) ® £ ) ^ ^ - J ^ ( C ) ® ( J ^ ( ^ ) ® ^ ( £ ) )
ffir((C®D)®E) - i ^ ( C ® £ ) ® J ^ ( £ ) ^ ^ ( ^ ( C ) ® ^ ( Z ) ) ) ® ^ ( £ ) . |
Such a funetor is called a m o n o i d a l f u n e t o r . If # and Q) are Symmetrie we require in addition the commutativity of
; s i ö
Let *// and be right ^-objects in Cat for a strictly monoidal category <ß. A morphism $ F \ J t - + J f is a funetor such that
&(M® C ) = ^{M)®Cy
where the identity is a natural transformation in M and C. In the general case of #-categories ^ and ./T for a monoidal category a (€-functor is a funetor & ' \ J l ^ J f together with a natural isomorphism
f : ^ ( M ® C ) ss ^ ( M ) ® C
such that the following diagrams commute:
# - ( M ® ( C ® £ ) ) * # - ( M ) ® ( C ® D )
J ^ ( ( M ® C ) ® i ) ) i J ? r ( M ® C ) ® Z ) ^ - ^ ( J z r ( M ) ® C ) ® J D
10 D
354 B. Pareigis
Finally we introduce natural transformations between monoidal functors, resp. between ^-functors. Let J 5 " : * ^ - ^ and be monoidal functors. A natural transformation ^ r J 5 " - » ^ is called a m o n o i d a l t r a n s f o r m a t i o n i f
^(C®D) - ^ ( C ) ® ^ ( 2 > ) i 1 \ x®x
<g(C®D) - i Öf(C)®»(Z)) and
commute. A natural transformation i j / : ^ - * ^ between ^-functors ! F \ M - + J f and < & \ J t ^ J f is called a %>-morphism i f
« F ( M ® C ) - i J**(M)®C I | ^ ® C
^ ( M ® C ) - i 0 ( A f ) ® C commutes. Now we can introduce the notion of a right ^4-object in a right ^-category ^ , where A is a monoid in C. A n object together with vM\M®A-+M is an A - o b j e c t i f the diagrams
M ® 0 4 ® v 4 ) 4 ( M ® ^ ) ® / 4 ^ - ' l M ® ^ 1 M®jj,A \ vM
M®A ^ M and
Non-additive ring and module theory II. 355
commute. It is clear how A - m o r p h i s m s are to be defined. Thus we get a category of y4-objects 3 ) A .
4.1. Lemma L e t 2 F \ Q ) ^ $ ) ' be a ^ - f u n e t o r . Then & induces a f u n e t o r
PROOF. Let ( M , v M ) £ ^ . We define ^ A ( M , v A f ) : = ( e ^ r (M), & ' ( v M ) ) where & ' ( v M ) is the morphism &(M)®A^^{M®Ä) H Z * * l & ( M ) . The diagrams
£(M)® CAeA) = ( ?'(M}© A)® A
Iii v. "(
HC sA)«s A
^ sCA® A))= >t(>fsA)«A)
and
AM;
commute. Similarly if f . M ^ N is a morphism in ^ then is in Q ) A since
^ ( M ) ® ^ ^ ( / ) 0 i t ^ ( j V ) ( g ) ^ \\l \\l
& ( M ) . <F(AT)
commutes. So define ^ ( / ) : = ^ " ( / ) in ^ and we get a funetor 3 F A \ 3 i A - + 2 ' A . This proof shows already why we had to require certain coherence conditions
for the definition of ^-functors. The most important property of ^-functors which is very similar to properties of exact functors will be discussed later on. First we need a few additional properties of ,4-objects in (€.
By Theorem 2.2 and the remark following 2.3 Lemma 3 of [12] the following is a difference cokernel of a contractible pair in ^
A ® ( A ® M ) ^ l L % A ® M A®vM
for each ^-object M in <€. In fact only the contraction morphism (rj<g)(A<g)M))' A-1:A<g>M-»A<g)(A<g>M) is in the morphisms of the above sequence are even in A
c€. So we have a difference cokernel of a ( ^ : ^ - ^ ) - c o n -tractible pair in
10*
356 B. Pareigis
4.2. Theorem. L e t & be a c o v a r i a n t f u n e t o r . E q u i v a l e n t a r e a) ! F i s a ^ - f u n e t o r a n d ! F preserves difference cokemels of % - c o n t r a c t i b l e
p a i r s . b) T h e r e i s a B — A - b i o b j e c t Q w h i c h i s A - c o f l a t ( w h i c h i s u n i q u e up t o a n i s o m o r -
p h i s m ) such t h a t 3F^Q®A as f u n c t o r s f r o t n A%> t o
PROOF. Assume that a) holds. Then we have a difference cokernel
& ( A ® ( A ®M)) =t &(A ®M) & ( M ) .
Since is a ^-funetor we get a difference cokernel
By definition of ^(A)®AM in §3 we get a natural isomorphism ßr(M)^^r(A)®AM with the ^-^-b iobjec t ßr(A) = Q. If Q'®A = 2F then Q ® AA^Q® A A hence Q ~ Q as B—y4-biobjects.
Conversely assume that b) holds. The following commutative diagram in-dicates that Q ® A : A ( & - * B & is a ^-funetor:
(Q®(A®M))®X^ (Q®M)®X (Q®AM)®X IR Wl II?
ß <g> (,4 <g> (M<g> X)) =t ß ® ( M ® - ß ® x ( M ®
The verification of the coherence diagrams for ^-functors is left to the reader. Now let
be a difference cokernel of a ^-contractible pair in $ with contraction g : N ^ M in # and section k : P - + N in # such that f 0 g = i d N , f g f ^ — f g f , h k — i d P and k h — f g . Now ß ® preserves the whole Situation, so we get a commutative diagram
Q® N
Q® Pi
Q® fe
i d Q<8N
Q® M
Q « f i
Q® N
Q® N
Q ® h
Non-additive ring and module theory IT. 357
This implies that
Q ®h
is a difference kokernel of a ^/-contractible pair [12, 2.3]. A similar diagram is obtained by tensoring with Q®A on the left. This induces a commutative diagram.
(Q®A)®M =t (Q®A)®N - (Q®A)®P U 11 \\
Q®M =* Q®N - Q®P \ \ \
Q®AM =* Q®AN - Q®AP
where all rows and columns are difference cokernels in due to the fact that / 0 , / i and Ii are y4-morphisms and that colimits commute with colimits. Hence Q<g>A :Aß-+Bß preserves difference cokernels of ^/-contractible pairs.
It seems that the strong property of being a ^-funetor rarely occurs. But the following theorem shows that this property is closely related to inner hom functors. Let us first consider the case <ß=Z — Mod . Let B
(ß have a right adjoint Then $F and ^ preserve colimits resp. limits hence they are additive
functors. Now the natural bijection
B # ( ^ ( A / ) , TV) £s A < g ( M , 9 ( N ) ) is composed of
and
Both maps are homomorphisms of abelian groups since & is an additive funetor and < P M : M ^ ^ ( M ) is in A
cß. Thus a pair of adjoint functors between A%> and B
(ß with <ß=Z — Mod is automatically such that not only the morphism sets $ { $ F ( M \ N ) a n ( i A ^ i ^ ^ i N ) ) are isomorphic in the category of sets but even the inner hom functors B [ ! F ( M \ N] and A [ M , &(N)] are isomorphic in cß=Z — M o d . Unfortunately this is not true in the general case. However the following theorem holds
4.3. Theorem, a) L e t (ß be a c l o s e d m o n o i d a l c a t e g o r y a n d ! F a n d ^'.ßß-^/ß be f u n c t o r s . $F i s left a d j o i n t t o <§ a n d a ^ - f u n e t o r if a n d o n l y if t h e r e i s a n a t u r a l i s o m o r p h i s m
B [ < F ( M ) , N ] ~ A [ M , $ ( N ) ]
f o r a l l M£A<ß a n d N£jß. b) L e t (ß be a c o c l o s e d m o n o i d a l c a t e g o r y a n d S F \ A m - + $ l a n d ^'.jfß-* A*ß
be f u n c t o r s . i s r i g h t a d j o i n t t o & a n d a <ß-functor if a n d o n l y if t h e r e i s a n a t u r a l i s o m o r p h i s m
, < * W ) , N ) ~ A ( M , $ ( N ) )
f o r a l l M£A<g a n d NeBW.
358 B. Pareigis
PROOF. First assume that there is a natural isomorphism
Then 9? is left adjoint to ^ by N)~BV(&(M)®I, N ) ~ B [ J ^ ( M ) , T V ] ) ^ ( / , Ä Cal l this
isomorphism cp. It is clear that <p is natural in M and TV.
Now define { : &{M®C)^^(M)®C by B W , N ) as B # ( ^ ( M ) ® C , W)ss ~ t f ( C , tf])~tf(C, ^[Af, ^ ( T V ) ] ) - ^ ( M ® C , ^ ( T V ) ) - B ^ ( M ® C ) , TV). Again £ is clearly a natural transformation in M and C.
To check the two properties for a ^-funetor denote by
* A ' ^ ( M ® C, M O - <f (C, TV'])
<AB : B#(TV® C, TV') - V ( C , B[TV, TV'])
the adjointness isomorphisms of 3.10. Then the diagram
A€CM, (N))
M M N ) ] )
commutes, the outer hexagon by definition of <p, the lower pentagon by definition of the upper right hand quadrangle since (p is a natural transformation and the Upper left hand triangle since all morphisms of the diagram are isomorphisms. Hence we get a commutative diagram in B*ß
TW® I
For the second more complicated diagram for ^-functors observe first from § 3 that we have natural isomorphisms
A[M® C, M ' \ a [C, A [ M , M ' ] \ and B[TV® C, TV'] - [C, B [ N , TV']]
and that the diagram
* D. . [ M. M'JJ = . [ M® (C<$ D) . M'] = , [(M®C) <s> D .
Non-additive ring and module theory II. 359
and the corresponding diagram with respect to #ß commute. Thus we get a commutative diagram
A[M®(C®D\ %{N)} ^A[(M®C)® A 9 { N ) ] = [ D , A[M®C,$(N)]]
[C® D , A [ M , 9 ( N ) ] ] * [ A [ C , A [ M , $ ( N ) ] ] ] \\l IK
[C®D, B [ P { M ) , N]] * [ A [ C , B [ ^ ( M ) , N ] J \
B[^(M)®(C®D),N] as B[(&r(M)®C)®D, N ] as [ D , B[3?(M)® C, N ] ] .
Observe now the following diagram
A C M ® ( C « D ) , ^ ( N ) ] - [ C ® D ; A [ M , ^ ( N > ] ] — ^ [ C a D , ß [ ^ M ) ; N J ]
f " \ ^ ^ t
, C ?( 3), Nj
gC'^CM) ® ( C ® D ) ,N]
[ 3, N]
B CrCM)®C)®D,N]
I [ D . CKM®C) ; N ] ]—TD , R[?CM)©C,N]]
CD, A [ M ® C , ^ . ( N ) ] ] LÖ . [C, B[;r (M) ,Nj ] ]
I [ 3 , [ C . A [ N , ^ ( N ) ] ] ]
The outer frame commutes, since the previous diagram commutes. The left quadrangle is commutative since <P is a natural transformation, the
right hand Square since \j/B is a natural transformation, the three outer pentagons by the definition of f. Since all morphisms are isomorphisms, the inner pentagon commutes also. Hence we get
&r(M®(C®D))- #"(M)®(C<g>£>)
J^((M<g)C)(g)£>) i - ^ ( A f ® C ) ® D - ^ £ . ( ^ " ( i l f ) ® C ) ® D
commutative. This proves one direction of part a,) of the theorem. Conversely, let be a ^-funetor and left adjoint to ^ . Then we have iso
morphisms
V ( C , B W { M ) , N}) s B < e ( P { M ) ®C,N)^ B<€{&(M®C\ N ) a
360 B. Pareigis
natural in Ce<gyM£A<g and N£B%. Hence B [ ^ ( M ) , N ] ^ A [ M , ^ ( N ) ] is a natural isomorphism.
The proof of part b ) of the theorem is essentially dual to the proof of part a ) and is left to the reader.
4.4. Corollary: L e t & : a n d : $ be f u n c t o r s such t h a t t h e r e i s a n a t u r a l i s o m o r p h i s m B [ ^ ( M ) , N] ~ A [ M , &(N)]. Then t h e r e i s a n A - c o f l a t o b j e c t Q ^ B ^ A ( u n i q u e up t o a i s o m o r p h i s m ) such t h a t
n a t u r a l i n M r e s p . N .
Proof: Since SF is a ^-funetor and preserves colimits, Theorem 4.2. implies 3F{M)^Q®AM. By Proposition 3.11 we also get 9 ( N ) ^ B [ Q 9 N ] .
If # = Z - M o d we know in the Situation of Theorem 4.3 that J 5" is an additive funetor. This holds more generally.
4.5. Proposition: L e t cß=K-Mod w i t h the u s u a l t e n s o r - p r o d u e t . L e t J5* : /ß-^ßß be a ^ - f u n e t o r . Then i s K - a d d i t i v e , i.e. f o r a l l f g£/ß(M9 N ) , x£K we haue
PROOF. First show that there is a natural isomorphism J r ( M © M ) ^ J 5 r ( M ) 0 0 & ( M ) (natural in M ^ A ^ ) such that
for /=1, 2 commute, where q{ are the z-th injections and p t are the i - t h projections of the direct sum. The isomorphism is given by &(M®M)2*&(M®(K®K))^ ^^(M)(g)(K®K)^^r(M)®^r(M). It is clearly natural in M£/ß. The commutativity of the first diagram follows from
#"(M) as Ö ® A M a n d 9 ( N ) as B [ ß , N ]
^(/+g) = ^(/) + #-(g) a n d & ( x f ) = x & ( f ) .
The other diagram follows similarly. The diagrams (*) imply immediately the commutativity of
and
Non-additive ring and module theory II. 361
Furthermore they imply the commutativity of
& ( M © M ) * «^( TV© N )
Hence
n M ) m — m
f ( / i @ M ) ^Cf®&„ $(N®N) A \ 7
commutes, where the first horizontal arrow is ^ ( f + g ) = & ' ( f ) + & r ( g ) . To show that ^ is AT-linear, observe that for f t ^ i M , N ) and x£K the diagram
\\l \\l M - N
commutes. Hence the diagram
£({) ® %
R H ) - F W
commutes, where the lower horizontal arrow is ^ r ( x f ) = x ^ r ( f ) .
Bibliography
B. PAREIGIS, Non-additive Ring and Module Theory I. General Theory of Monoids. Pub/. M a t h . (Debrecen), this vol. 189—204.
(Receiued November 20, 1975.)
