GENERALIZED EQUIVALENCE OF MATRICES OVER PREFER DOMAINS

Two m x n matrices A,B over a commutative ring R are equivalent i,.ve,-tible nmtrices P, O over R with B PAQ. While any m x n matrix over a principle ideal dota.i, ca, be diagonalized, the same is not true for Dedekind domains. The first author and T..I. Ford ittroduced a coarser equivalence relation on matrices called homotopy and showed any x mtrix over a. Dedekind domain is homotopic to a direct stun of x 2 matrices. In this article wc giw, necessary and sufficient conditions on a Prefer domain that any m x n matrix be homotolfic to a. direct sum of x 2 matrices. l(cy Words and Phrases: Priifer domain, Progenerator module, Bezout domain, matrix equivalence 1980 Subject classification codes 13F05, 13C10, 15A33.

Let M,N be finitely generated projective faithful modules (progenerators) over a. co,nmuta.tivcrig R.An R homomorphism h M N is called image split in case h(M) is a faithful R-direct summand of N. If I :M N and g:P Q are homomorphisms of R progenerators then are said to be homotopic if there are image split homomorphisms h A B and isomorphisms , making the commuting diagram of R-modules MA I N@B PC O D If I P-'g for isomorphisms p,v then I and g are homotopic (where R A B C O h. ln).Thus equivalent homomorphisms are homotopic but not conversely.The notion of homotopy of homomorphisms w introduced in [4] to remove most of the obstruction observed by L. Levy in [13] to diagonalization of matrix transformations under equivalence over Dedekind domains.Summarizing some of the results in [4], homotopy is an equivalence relation on homomorphisms of progenerator modules and tensor product of homomorphis induc a multiplication on homotopy classes which turns this set of cls into a monoid denoted M(R).Each homotopy class is represcnted by at let one matrix transformation, and if R is a Dedekind domMn by a matrix transforma.tiowhich is a direct sum of x 2 matrices, a matrix of the form a b 0 0 0 0 a. be m x 2m 666 F. DeMEYER AND H. KAKAKHAIL Moreover, if Ij ajR + bR then I D 1._, D D If R is a discrete valuation ritg lwn .(PN[a.]. the mouoid of primitive polynofia,ls with coefficients iu N {0,1,2,..} togctltcr wil, l 0-1)olytomial.If R is a Dedekind domai tlen M(R) is naturally isomorphic to (],,,w,st,m ( atttl tltis isomorphism gives an isomorl)hisn between M(R) and primitive polynomials over N ideterminates indexed by MazSpec(R).
q'le purpose of this paper is to determine the extent to which these results can be gctwralizcd to arbitrary doma.ins.In fact, they come close to characterizing Dedekind domains.We first olsct tltat if li' is a commutative ring containing a maximal ideal P such that dinn/t.(P/ P") > 2 tlt(;t, is a honotopy class in M(R) which contains no matrix transformation which is a direct sutn of x tmtriccs.Thus, if R is a Noetherian domain and every homotopy class in 3A(R) contains a ml, rix wlich is a direct sum of x 2 matrices then direr < 1.The inclusion map from a domain R to its itcgral closure R induces a monoid homomorphism .1(/)-(/)which was steadied in [6].llcrc relax l,]e Noetherian condition and study .(/)for Prfifer domains.If R is a Priifer doma,i o1" Krll ditnension or if R is a Priifer domain of finite character (each nonzero element, of R is i otly finitely many maximal ideals) we show every class in .A(R) contains a represet|ting natrix wlicl is a direct sum of x 2 matrices.If R is any valuation domain with value gt'oup (:; < (R, +) (;+ is 1,]e nonoid of nonnegative elements of G form the monoid PN(G+) of "primitive l)olynomials" :.,,,:+ o.x. with a N, ahnost all a 0 and 9cd{%lgeG + 1.We show j4(R) PN(G+).After givi,g a sliglt generalization of L. Levy's "Separated DivisorTheorem" for matrices over Dedekittd donains [13], we can show for Priifer domains that .I(R) is naturally isomorphic to t.,nt.sr.(n).,'vl(Rt,)it" and o.ly if R is of finite character and the valuation rings at the maximal ideals of R arc pairwisc independent.The principal examples of Prfifer domains of finite character whose valtation rings at m,ximal ideals are pairwise independent are Dedekind domains and valuation donmins.
Part of this paper appeared in the first author's Ph.D. dissertation written at Colorado Uniw:rsity.This paper was completed while the second author was a visitor at Florida Atlantic University.He wishes to thank department chairman Jim Brewer for his hospitality.We would also like to thank L. Levy for his help with the proof of the generalized Separated Divisor Theorem."2.SEPARATED DIVISOR THEOREM PROPOSITION 1.Let R be a conunutative ring containirg a maxinal ideal p wil.l dimn/t,(P/P') > 1.Then there is a matrix transformation over R which is not homotopic to a di- rect sum of x 9. matrix transformations.PROOF: Let .R S be a homomorphism of commutative rings, so S is an R-algebra.Then itduces a monoid homomorphism M(q)'M(R)-M(S) by M()(lf[)= [I(R)Yl, where if y Homn(N,U.) the 13" Hotns(SC)U,,S(R)U,.) (Theorem of [4]).Since each class in M(S)is represented by a matrix transformation, if 0 is an epimorphism then M() is an epimorphism.If is an epimorphism and if every class in M(R) is represented by a matrix which is a direct sum of 2 matrices, then ew.ry class i M(S) is represented by a matrix which is a direct sum of x :2 matrices.Thus, it suffices to check tle conclusion of the proposition for a homomorphic image of R. Let {ax + P,a+ P-} u {c,, + P"},t be a basis for P/P" over RIP.Let J be the ideal in/ generated by P and {a,},,t.The ring S R/J is a local ring with maximal ideal M P + J/J.Moreover, M (0) and dinslM(M/M ) 2. Let a,a.., M be linearly independent over S]M We check the matrix aa a:/ is not homotopic over S to ally 0 ax i.at,'ix of the form H 0 .., Since S is local, with respect to a suitable basis choice, each image split homomorphism has a ,natrix representation of the form diag(l,.., 1,0,..,0) (Proposition 3(9) of [4]).We need to check F= aO a a ' 2 ] diag(1 1,0, ,0)= m x 2m is ,lot equivalent over S to H. View F as the relation matrix of the factor module SO")/Lr where l,r is the submodule of S (2") generated by the rows of F. Then S(2)/Lr is isomorphic to a direct sul, of modules of the form A S S < (a,a2),(O, al) > together with 0-summands.A (lit'oct ('alcttlation shows the S-endomorphisms of S, S leaving < (a,a2),(O,a) > invariant are givett "'atric"s t'the frn a f t a+m'n' 'vhere 'n''''' M'' S" That is' Ends(A) is a hmm"l)"ic imgo of the ring of these natrices.A direct but msy calculation shows that if +,nJ which is nilpotent we s Ends(A) has no idempotents other than 0 and so A is an indecomposable S-module.In the same way view G as the relation matrix of the factor ,odue S"'/L; where La is the submodule of S generated by the rows of G. Then S'"I/La is ismnorphic to a. direct sum of modules of the form B, S S/< (a,,B,) >.An easy calculaton shows d,sh(M" A) , dmsm(M.B,) a, and dimsl(A/MA) 2 dimsl(B,/MB,)(1 m).If A then A/MA/ML B,/MB, so L ML so by Nekaya's lemma L (0).In this ce A B, which is impossible by the first dimension count above.Thus A is an indecomposable S-module which is not a direct summand of any B,.By the Krull-Schdt Threm Somalia S(/ and F,G cannot be equivalent matrices over S. REMARK" If R is noetherian then dim R supes,e()dimle(P/P ) so if R is noetherian, Proposition implies that if every matrix over R is homotopic to a direr sum of x 2 matrices then dim R 1.This may not be the case when R is not noetherian the next result shows.
Let K denote a field and v a vMuation on K with vMue group c (R, +).Let R be the valuation ring corrponding to v. Since R is an elementary divisor ring [9], each mx n matrix over R is equivalent to a diagonM matrix diag{d,...,d} with v(d,) v(+) when d,+ 0 and d 0 impli d 0 for R j.The "elementary divisors" d d uniquely determine the equivMence cls.In this case, following [4], we can explicitly determine the monoid of homotopy classes.Let G + l and N(a+) {() = n,'n,e N,,e a+}, where N is the set of nonnegative integers.The, N(G+) is a nultiplicative monoid with multiplication induced from the equation , '*,.For (.),b(x) N(G+) say () b() if there exists positive integers r and with r() sb().It is esy to check that is a congruence on N(a+).Let PN(a+) N(a+)/ Then PN(G+) is a monoid which can be identified with the primitive polynomials in with exponents from G+.Let I() be the congruence cls in PN(a+) represented by ,() N(a+).To the homotopy cls in (R) represented F. DeMEYER AND H. KALHAIL by the lxl matrix transformation (d) over R we can assign the congruence class Izv(dl in PN[s.].Our text result is that this assignment extends to an isomorphism.PR.OPOSITION 2. If R is a valuation ring corresponding to a valuation v on a field K witl value group G then M(R) PN(G+).PROOF: Lemma 2 and Proposition 3 (l) of [4] imply any 0 Ihl, M(R) contains an x r matrix transformation diag{d,,...,d} where d, 0 and v(d,) v(d,+) for all i.Let (R) PN(G+) by (Ihl) = x(a').We only check is well defined, then the rest of the argument is routine.If Idiag{d, a}l Id-a{Y, Y,}I in M(R) with v(I) v(I+) for all j then by Proposition 3 (9) 0} is equivalent to diag{y y,}@diag{1 1,0 0}.Let be the entries in diag{d dr} with pairwise distinct valuations.By uniqueness of invariant fa.ctors i, a, elementary divisor ring, the entries in diag{I I, with distinct valuations are f', y.where ,,{f) ,,(d) whe,, we order d,f so v(a',)< o(d',+) and o(f) < ,,(/+) for all i.Let , ., #{djlv(dj)= ,,(d',), S J S "} and .,,#{Yl"(/,) v(/;)l S J S s}.Then @(Ihl) IE=t ",zta")l and @(Ihl) I= Moreover, by uniqueness of invariant factors, pr, qs, S 5 k so I,= r,a;)l I,= s,x"l;l i PN[x'].'l'hus 4 is well defined.
The following is needed to prove a generalized separated divisor threm.Undefined termiology can be tbund in [7].
LEMMA 3. Let R denote a Prfifer domain of finite character whose valuation rings at ma.ximal ideals are pairwise independent.If 0 # L is an ideal in R then there is a factorization L =t L, where each L, is contained in exactly one maximal ideal P, of R and P, # P if # ).
PROOF: Since 0 # L and R has finite character, L is contained in only finitely many naxinml ideals P,,...,P of R. Let L, LRp, flR(1 G k). Theorem 4.10 of [7] implies L flptaasp,c(n)(LReR). Since LRp Rp if L P we have L ,I(LRp, R) O,IL, We always have L, C P, Let v,,v be valuations corresponding to P,,P respectively.Since these valuations are pairwise independent, Theorem 22.9 (2) of [7] implies that for each 0 # z L there is an a Re, Rp, with v,(a) and u(a) 0. If S R-P, O then an elementary exercise gives, S -R Re, Rp, so after clearing the denominator we can sume a R. Thus a LRp, R L, but a P so each Li is contained exactly one maximal ideal of R. Morver, L, + L R whenever # j since L, + L# is contained in no maximal ideals of R. Thus L =L, H, L,.To check uniquens sume L H= L where each L' is contained in exactly one maximal ideal P of R and P # P'if j # q.From the above w and after relabeling we can sume P Pj.Now LRp, (= Lj)Re, LjRp, so if, Re, L, Rp,.Since L LRp,R it follows that L', C L, But Rp, L', Rp, L, and RpL', RpL, Rp if P is a maximal ideal of R not equal to so Rp Li/L' 0 for all maximal ideals P of R so L L' i.
Following [13], the ideals L, in Lemma 3 are called the separated divisors of L. The separated divisors Div{d,}=t of a finite sequence of ideals in R is the collection, counting multiplicity, of all the sepa.rateddivisors of the individual ideals J,.
For convenience we list some definitions and a result we need from [2].A Prfifer domain R with quotient field K is said to satisfy the Invariant Factor Threm if for any finitely generated submodule M of %"} there exist simultaneous decompositions of R ") and M R " ' ...-t Jz ... M Elz q E,._Iz,._q) E,.z,.
where , K, the J, are invertible fractional ideals of R, the F-,i are invertible integral ideals of R and E, C E,+ for 1,2 1.A Prefer domain R has the Steinitz property if for fractional ideals and J, q J _ R 1.I.A Prfifer domain R has the 11/2 generator property in case fi)r any fi'aclioal itleal landany0#zt lthereisayt lwitl l=+Ry.Let R be a Prfifer dotnin and A an m x mtrix over R of rank r.Let Ma be the submodule of R( " genera by the rows of A and let Sa I. It'R is of finite character or direr then there exist invertible ideals E E of R witl ) and invertible fractional ideals , O+ 0,, such that = R/E, + J+ +... + d. if ," < ,, =, R/E, if ,.= , wlered7 ==E, ifr=,n, =JRifr=Oand J,Rif,'=n. [[.I["/t is of finite character with pairwise independent valuation rings at maxitnal ideals and A,A' are two ,,x, matrices over R of rank then A is equivalent to A' if and only if Dv{E,}'" Dio{k.'}'=PROOF: I. Let M be any finitely genert submodule of Rt".Since R satisfies the hva.riat Factor Threm there exist simultanus decompositions of R" and M, M EI E__ E,.Jr,.
with a-, K quotient field of R, the J, invertible fractional ideals of R and E, invertible integral itleals of R such that , c E,+ < < r-1.Since R satisfies the 1 generator property, Proposition of [2] implies Jr/ErJr R/Er.Hence (where d+ ...d ,, do not occur if n).Here r is the rank of M which is the rank of A if m is generated by the rows of A. This prov SA h the decomposition given in I.If then so E .. E,__ EJ "').The Steinitz Property and ccellation imply = E,.I,.R and Jy =l E,.In the same way, if r 0 then R") d ... J,, and J...J, J,+ ...J, R.
To complete the proof of II we need to check that if Sa Sa, then A is equivalent to A'.Our problem is to find isomorphisms , 0, a making the commuting diagram Rm Rm) Sa 0 o A' Snce R 8 re, beorem 1.8 o [9 mphe8 .? 6ere ? 8poectve and If SA P is the projection let p be a splitting map so R(" p(P)+Q and ker ,1C Q.It the same way S a, P'+T', R () p'(P')+Q' and ker 1' C Q'. Since Sa Sa, there are isomorl)hists a: !'--!" aud 7'+ 7".By Proposition of [12], T is a direct sum of cyclic modules R/L for ideals 0 L C 1.
By lean,ha 3 we can let {L,} be the set of separated divisors of L. The Chinese Retnain(h'r implies R/L +R/L,.Since L, is contained in only one maximal ideal of R, R/L, is local tbr all i.
Since Q/ker o 7", Theorem 1.6 of [11] implies there is a simultanus depositiou of Q an(I ker 0 In the same way there is a simultanus decomposition of Q' and ker 0'.Thus the given isomorpltis T 7" extends to an isomorphism :Q Q' such that +(ker 0) ker 0'.This gives isom<>rpliss + and + p'a + making the commutative diagram SA 0 Sillc(' 07(ker 0) ker,l' we have the exact R( 'A' IrnageA'---.O ald :, is the lt-honomorphism given since R '' is free.This completes the proof of the Separated Divisor Theorem.
3. THE MONOID OF HOMOTOPY CLASSES LEMMA 5. Let R be a Pr/ifer domain of finite character or Krull dimension < 1.
(1) If 0 # Ill .(R)then there exists g such that Ill Igl in sM(R) and coker(g) is a torsio R-module.
(2) Let Istl, lgle ./I(R}with coker(g) and coker(st) torsion R-modules.If M g g P Q theu I/I Igl in .X4(R) if and only if there exists R-progenerators K,L such that g (R) If Q 63 L and coker(f) (R) K coke,'(g) (R) L.

PROOF:
The proof now follows "mutatis mundantis" as the proof of lemma 6 of [4].
Following page 393 of [4] a description of Yt4(R) in terms of ideals of R can be given now.Consider by p(M, R(')) Jil where M R (') is the inclusion map.Note that by Lemma 5 p(N, R(")) if and only if there exist R-progenerators K and L such that Thus t) induces an equivalence relation on A as follows: Let .4be the set of equivalence classes of A. Then the product on A induces the multiplication o, .4,turning it into a commutative monoid with identity the class containing (R,R).LEMMA 6.Let R be a Prfifer domain of finite character or Krull dimension < 1, then tlw nlap " .4 .(R) induced by p is an isomorphism.PROOF: Same as Proposition 7 of [4].THEOREM 7. If R is a Priifer domain of finite character or Krull dimension < then every homomorphism of R-progenerators is homotopic to a matrix transformation of the form where if/ =(a,b) then I 3l+(l<j<n-1).
PROOF: The proof of Theorem 7 now follows exactly as on pages 391-394 of [4].LEMMA 8. Let R denote an integral domain.Then the following are equivalent.
Each nonzero element of R is contained in only finitely many maximal ideals of R.
2) If N is a finitely generated submodule of R(") then Re (R) N is a direct summand of R ") for almost all PeMaxSpec(R).
(3) For each exact sequence 0 N Q m 0 of finitely generated R-modules with Q projective' the associated sequence 0 Rt,(R) N R,(R)Q Rp(R) M 0 is split exact for almost all PeMa:cSpec( /i') PROOF: The equivalence of 2 and 3 follows easily since every finitely generated projective naodule is a direct summand of a free module of finite rank.To see 2 let 0 a R. Then (a) is a submodule of R so (R) is either 0 or a unit in Re for almost all PMaxSpec(R).If R is an integral domain then (R) a 0 in Re so a P for almost all PeMa:Spee(R).To show 2 let R(") Rx ...$R ,,, and let N Rn +... + Rn, with 0 n, R(n}(1 _< _< k).Let n a +... + a,:, where we can assume a 0.Over a'(R we have a[n,x,. xn is a basis for (a-(R) ") and a-{n,n:,..,n: generates a'N so replace x and n by a{n over ajaR.Then a,,...,,, is a basis for (a'R) n and z,n.generates a? N. Write n. bz + b.z + + b,x,.Since ze a[N we can replace n by b.z +... + where be 0 or ,e a'[Rn a'{R:.If b. 0 then over ba-(R we have ,bn.,:z,, is a basis for (b.a-[R) ") and z,btn2,nz nt, generates baN so replace and n by bn over b.a'R.
Afler finitely many steps we can find an element c R with c-N a direct summand of/i5").Since c is in only finitely many naaxima.lideals of R by 1, Re (R) N is a direct summand of R '*) for almost all PMa:Spec(R).
If R is an integral domain of finite character then a nonzero homomorphism m N of R- progenerators induces an image split homomorphism R,(R)MX-,IRt,(R)N for almost all PeMaxSpec(R)by 3 of Lemma 8.In this case the inclusions R Re induce a natural map :M(R) LEMMA 9.If R is a Prfifer domain of finite character and M(R) ,,M,,s,n)M(R,) is induced from the inclusions R Rp then is a monomorphism.PROOF: Let 0 Ill 0 Igl be in M(R) and assume (I/'1) (Igl).Then [1 (R) II l1 (R)   for all PMaxSpec(R).By Theorem 7, 0 Is'l determines a descending sequence 1 I.
::) 1,, of ideals in R with 0 I (a,b) and 0 Igl determines J or,,, with 0 or, (c,,d,).Under j R,,) the isomorphism # of Lemma 6, (']=I R") corresponds to Ill and (9= , corresponds to I.ql- We show (']=I,R ') (or,R) in A Let P be a maximal ideal for which II (R)  is split.In A(Rp) the split class is represented by (Re,Re) and projective modules are free over Re so (R(')/=t I) tt) (R) Re (0) which implies I c P for j 5 n.In the same way y c P so lRp Rp JRe for all i,j whenever 11 fJ Jl l is split over Re.Consider the finite set (si,ce R h finite character) S {PMazSpec(R)II1 fl Ila.I} {PMazSc(R)II1 @ l # Iln.I}.Since 17= e,,ng'l IZ neJ,nl in A(Rp) for eh P S there are positive integers se,te with spn tpm and [Rp/Rpb]t'" [Rp/RpJ] ).Let He,ssp and p,stp.Then for each ,=1 [RP/Jk] (t)" The ideals Rt, I and ReJk in this decomposition are uniquely determin as sets (ii pg 260 of [12]).This means the list of ideals RpIt Re I and RJ RFJ... where It I and J J are the sa,c for each p S. Therefore, for each j,t, P,Ms,,n)(J Re R)   for corresponding k, w We have shown [t(R/I)] ' [=(R/J)] so (R),=6) (R",=J) in A and is a ,onomorphism.
We need two ey lemm about valuations on Prefer domains.
IEMMA 10.Let R be a Prefer domain of finite character whose valuations at ma.ximal ideals re pairwise independent.Let p p, be a finite set of maximal ideals of R, let G, be the value group of Re, and let 0 < a, G,(1 n).Then there exists a finitely generated ideal of R such that IRe, {o Re, lve,() 9i} and the only maximal ideals of R containing are PROOF: Since the valuations at the maximal ideals of R are pirwise independent, Theorem 22.9 of 7 implies there exists an x Re, ... Re. such that ve,(x) ,(1 n).As we observed in the proof of Lemma 3, we can choose z R. Sin R h finite charter we can let O Q,, be all the maximal ideals of R distinct from {Pi},% such that r O(1 S j m).Again, Theorem 22.9 of [7] implies there are u, R with ve,(,) , and v,(,) 0 for all j i,k.Let I (x, u).Then is finitely generated, IRe, zRe, { Re, lve,(a) E 9,} and the only maximal ideals of R containing re P, ., Pn.
LEMMA 11.Let v,v be valuations on field K with value groups Gt,G2 and valuation rings V, V respectively.If for ch pair (9,) G x G with 0 S and 0 S 9 there is an and v(r)= 9 then the valuation rings v, v are independent.
PROOF: (S 9, pg 289 of [7]).THEOREM 12. Let R be a Prffer domain.The inclusion maps R Re induce n isomorphism '(R) e,,s,,n)(Re) if and only if R is of finite character and the valuation rings at the maximal ideals of R are pairwise independent.PROOF: Assume R is a Prffer domain of finite character.Lemma 9 giv is a monomorphism.We check that if in ddition the valuation rings at the maximal ideals of R are pairwise independent then is an epimorphism.Let (lgel)e,,s,,<n) be an element of e,,s,,n)(Re).Then ae is an inaage split map for all but finitely many maximal ideMs p p of R. Each I.1 can be reprented by a diagonal matrix (Proposition 2).By tensoring these matric with identity matrices of appropriate sizes we can sume each 19,1 is reprented by a diagonM m x n matrix.Let 19e, be represented by

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Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: nd fl =,, =m'=O-Sinceidempotents can belifted modulo nilpotent e,B S,m m' M o Ends(A) has kernel ideal and the natural homomorphism from e + m + ,,m,m'

PROPOSITION 4 .
If R is a Priifer domain of finite character or Krull dinmnsion tlc gent'ral ll satisfies the Inva.riantFactor Theorem, R has the Steinitz property and R h the 15 property PtOOF: [] SEPARATED DIVISOR THEOREM (Levy).