Jukl, Marek

Institute of Mathematics AS CR, v. v. i.

Acta Universitatis Palackianae Olomucensis. Facultas RerumNaturalium. MathematicaMarek JuklGalois triangle theory for direct summands of modulesActa Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 39 (2000), No.1, 67--71Persistent URL: http://dml.cz/dmlcz/120417Terms of use:© Palacký University Olomouc, Faculty of Science, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.cz7) Acta Univ. Palacki. Olomuc, Fac. rer. nat., / Mathematica 39 (2000) 67-71 Galois Triangle Theory for Direct Summands of Modules M A R E K JUKL Department of Algebra and Geometry, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic e-mail: jukl@aix.upol.cz (Received September 13, 1999) Abstract The purpose of this paper is to find an analogon of the Galois triangle theory for direct summands of modules over arbitrary unitary commuta-tive rings. Key words: Module, direct summand of module, ring of endomor-phisms, anihilator, isomorphism of ordered sets. 1991 Mathematics Subject Classification: 13C99, 06A15 Let an arbitrary unitary commutative ring A. be given and let M be an A-module. The aim of this paper is (by using the properties of projections) to find 1-1 correspondences between the ordered set of direct summands of M and the ordered set of left (right) principal ideal of the ring of endomorphisms of M generated by an idempotent element. The solution of this problem is well known f.e. in the case M is a vector space [1] or totally reducible module [3] (the set of direct summands of M is equal to the set of all submodules in this cases). Notat ion 1 1.1. By P we will denote the ring of endomorphisms on M i.e. P = E n d M , the composition of / , g G P we will define by (fg)(x) — g(f(x)). * Supported by the grant of the Palacky University 1999 "Rozvoj algebraickych metod v geometrii a uspofadanych mnozinach" G7 68 Mareit JUKL 1.2. By L(P) we will denote the set of the all left ideals of P generated by an idempotent element and by i?(P) the set of the all right ideals of P generated by an idempotent element. 1.3. By U(M) we will denote the set of the all direct summands of the A-module M. 1.4. For every S G U(M) let us denote N(5) = { / G P ; V x G S : / ( x ) = o } , Q(5) = { / G P ; V X G M : / ( X ) G S } . (Equivalent^, N(5 ) = {/ G P ; 5 C K e r / } , Q(S) = {/ G P ; I m / C S}). 1.5. For every J G R(P) let us denote K(J) = {x G M ; V / G J : f(x) = o} , L(J) = { / G P ; V 5 G J : / < ? = o}. 1.6. For every J G L(P) let us denote r H ( J ) = {x G M ; 3 / i , . . . , / r G J , 3 y i , . . . ,y P G M : x = ] T / ^ y * ) } , R ( J ) = { / G P ; V 5 G J : 5 / = o}. i = 1 R e m a r k 2 Every idempotent element of P is usually called a projection. It is well known (see [2]), that / G P is a projection if and only if there exist elements U, V G U(M) such that: ( l ) M = U©y, (2) VxG M , x = x r r + x y , x c / G U,xK G V : / (x ) = Xr/. Therefore / is called a projection M OHiO U parallely V. Clearly, I m / = U, / = V and / |U is an identity mapping. L e m m a 3 For any f,g G P , / is a projection, the following holds: K e r / C Kerg <=> fg — g. P r o o f Let K e r / C KerG. Let us consider an arbitrary element x of M . It may be written by x = xi + x2 , xi G I m / , x2 G K e r / . Then we have g(x) = g(xi)+g(x2) = g(x!) = g(f(x1)) = g(f(x)) = (fg)(x). The reverse implication is clear. L e m m a 4 For any / , a G P . / z's a projection, the following holds: Img C I m / ^=4> g/ = o. P r o o f Evidently, g/ = g implies Imo C I m / . Let Im a C Im / . As / | Im / = id we may write (fl/)(x) = / (g(x)) = g(x), for any x G M . Galois triangle theory for direct summands of modules 69 Proposit ion 5 For any submodule S G U(M) the following hold: (l)N(S)eR(P), (2) Q(5) e L(P) . Proof Firstly, let us prove the part (1). Let us consider an S G U(M). It follows from the definition of the operator N than N(S) is a right ideal of P . Since S G U(M) we have a T G U(M) such that M = S&T. Using remark 2 we obtain the existence of a projection / M onto T parallely S = K e r / . Let an arbitrary endomorphism g G N(S) be given. It yields S C Ker a which implies (by lemma 3) that g — fg. Thus N(S) C / P . Since the reverse inclusion is evident we have N(S) G I?(P). The part (2) may be proved by an analogical way (due to lemma 4). Proposit ion 6 If J G It(P) then K(J) G U(M). Proof Let / be the idempotent generator of J, J = / P . Using the definition of the operator K we have x G K(J) iff (/p)(x) = o for every PGP, which is equivalent to / ( x ) = o. It yields K e r / — K(J ) . As / is a projection the kernel of it is an element of U(M). Proposit ion 7 If J G L(P) then H(J) G U(M). Proof If we denote by / the idempotent generator of J then the submodule S = I m / belongs to U(M). Arbitrary x G H(J) may be written as follows (fi G J for any i — 1 , . . . , r ) : r r r x = Y2 h(yi)= H (ff./)(y.) = £ /(s^y.)). i = l i=l i=l which gives that H(J) C S. The reverse inclusion follows from the definition of H, immediately. It follows from the propositions 5, 6, 7 that it is possible to investigate the compositions K N , NK, HQ and QH . Proposi t ion 8 IfS G U(M) then the following hold: (1)K(N(S)) = S, (2) H (Q(5)) = S. Proof Firstly, let us prove the part (1). If S G U(M) then there exists a projection / such that K e r / = S and moreover N(S)) = / P (see the proof of proposition 5). Using this fact and the definition of K we may write K(N(S)) = K e r / = S (see the proof of 6). Now, let us prove the (2). 70 Marek JUKL It follows from the lemma 4 that for any S e U(M) we have a projection g such that Img = S and Q(S) — Pg. Since every element from H(Pg) may be written by r r x = XXfo)(y.) = £0(/.(yi)). L,...,/rGP it follows from this H(P#) = Img. It means H(Q(5) ) = S, consequently. P r o p o s i t i o n 9 (1) IfJe L(P) then Q(H(J)) = J. (2) IfJe .R(P) £/*en N(K(J)) = J. Proof Ad (1): Let / be an idempotent generator of J. Then H(J) = I m / (see the proof of 7). Respecting the fact g is an element of Q ( I m / ) iff Img C I m / we (according to 4) have g e P / = J. Ad (2): Let / be an idempotent generator of J. Then K(J) = K e r / (see the proof of 6). Using the lemma 3 we obtain that g e N(Ker / ) iff g e / P = J. P r o p o s i t i o n 10 (1) IfJe R(P) then L(J) = Q(K(J ) ) . (2) IfJe L(P) then R(J) = N(H(J ) ) . Proof Firstly, we prove the part (1). Let J = / P , Then K(J) = K e r / and therefore Q(K(J ) ) = Pg where g is a projection with Img = Kerf (see proofs of 6 and 8). Let us consider arbitrary elements q e Q(K(J ) ) , j e J which means that q — pg and j = / r , where p, r e P . For any x E M we may write (qi)(x) = r(f(g(p(x)) = o , since Img = K e r / . Thus Q(K(J)) C L(J) . If h e L(J) then hf = o. It implies that Imh C Ker / . Using the fact K e r / — Img and lemma 4 we obtain h — hg. It implies n £ P# = Q(K(J ) ) . The second part may be proved analogously. It is easy to derive that for every U,S e U(P), J, K C P hold (1) JCK^ K(J) D K(iv"),H(J) C H(iv"),R(J) D R(.ftT),L(J) D L(iv"), (2) U C S => N(U) D N(5) ,Q(U) C Q(5) . Now, if we consider operators N , K , Q , H , L , R a s mappings of correspond-ing ordered sets then the following theorem follows clearly from propositions 8, 9 and 10. Galois triangle theory for direct summands of modules 71 Theorem 11 (1) Operators N and K are mutually inverse antiisomorphisms of ordered sets (U(M),C) and (.R(P),C). (2) Operators Q and H are mutually inverse isomorphisms of ordered sets (U(M),C) and(L(P),C). (3) Operators L and R are mutually inverse antiisomorphisms of ordered sets (it(P),C) and(L(P),C). (4) The following diagram is commutative R ДP) ^ Д(P) U(M) References [1] Bаer, R.: Lineаr Algebrа аnd Projective Geometry. Izdatelstvo innostrannoj literatury, Moscow, 1955 (in Russiаn). [2] Bourbаki, N.: Algebre. Nauka, Moscow, 1966 (in Russiаn). [3] Mаchаlа, F.: Über Automorphismen eines Annullatorenverbandes gewisser teilringe im Endomorphismenring eines homogenen vollständing reduziblen Moduls. Actа Univ. Pаlаcki. Olomuc, Fаc rer. nаt., Mаth. 41 (1973), 15-26. 

English

Galois triangle theory for direct summands of modules

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/120417/ActaOlom_39-2000-1_5.pdf

Galois triangle theory for direct summands of modules

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library