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1 Volume 16 pp April 27 ON ROTH S PSEUDO EQUIVALENCE OVER RINGS RE HARTWIG AND PEDRO PATRICIO Abstract The pseudo-equivalence of a bloc lower triangular matrix T T ij overaregular ring its bloc diagonal matrix D(T )T ii is characterized in terms of suitable Roth consistency conditions The latter can in turn be expressed in terms of the solvability of certain matrix equations of the form T ii X YT jj U ij Key words Regular rings Roth equivalence AMS subject classifications 16A3 15A21 1 Introduction definitions Let R be a ring with unity 1 let R m n be the set of m n matrices over R shortenr n n to R n Throughout all our rings will have an identity An element a Ris said to be regular if a axa forsomex which is denoted by x a R is said to be regular if all of its elements are regular A reflexive inverse a is an element x such that axa a xax x We shall denote such an inverse of a by a + The sets of inner reflexive inverses of a if any will be respectively denoted by is a{1} a{1 2} Definition 11 m n Rare pseudo-equivalent m n provided there exist regular elements p q p q such that n pmq m p nq We may without loss of generality replace the inner inverses p q by reflexive inverses p + q + AringR is called (von Neumann) finite if ab 1 implies ba 1 it is called stably finite if R n is finite f or all n N AringR is called unit regular if for every a in R aua a for some unit u in R When p>qthen R p finite implies that R q is finite Matrices A B are said to be equivalent denoted by A B ifa PBQ for some invertible matrices P Q Liewise matrices A B are said to be pseudoequivalent denoted by A B ifa PBQ B P + BQ + for some square matrices P Q with reflexive inverses P + Q + If A A ij isablocmatrixoverr with A ij of size p i q j i j 1n then A with n istheleading principal bloc-submatrix A ij withi j 1 The trailing principal submatrix is given by  A ij withi j n +1n 1n Received by the editors 21 March 26 Accepted for publication 11 March 27 Hling Editor: Roger A Horn Mathematics Department NCSU Raleigh NC USA (hartwig@unityncsuedu) Departamento de Matemática Universidade do Minho Braga Portugal (pedro@mathuminhopt) Member of the research unit CMAT Centro de Matemática da Universidade do Minho Portugal 111

2 Volume 16 pp April RE Hartwig P Patrício ie A? A? Â n Definition 12 For a bloc matrix A A ij we define its diagonal as the bloc matrix D(A) diag(a 11 A nn ) We further set D (A) diag(a 11 A ) 2 Roth conditions Consider the matrix A M B D When the matrix equation DX YA B has a solution pair (X Y ) one checs that I A I A (21) Y I B D X I D ie M is equivalent to its bloc diagonal matrix N diag(a D) On the other h for regular matrices A D the consistency of DX YA B is equivalent to the condition (1 DD )B(1 A A) for some hence all D A Given consistency it was shown in 4 that over a regular ring the general solution to DX YA B is given by X D B + D ZA +(I DD )W Y (I DD )BA + Z (I DD )ZAA where W Z are arbitrary In 1952 WE Roth proved the converse of (21) for matrices over a field F 16 ie A (22) DX YA B has a solution pair if only if B D A D A ring R is said to have Roth s equivalence property if the equivalence (22) is valid for all matrices over R Roth s equivalence property was extended in 1 where it was shown that over a unit regular ring a b d a d dx ya b has a solution pair (x y) This result implies that such rings must be finite have Roth s equivalence property It was later extended to regular rings by Guralnic 6 who showed that over a regular ring Roth s equivalence property holds if only if R is stably finite In a parallel paper 7 Gustafson proved that a commutative ring also must have Roth s equivalence property We shall show that in Roth s equivalence property we may replace equivalence by pseudo-equivalence provided the diagonal blocs A i are regular R τ is finite for suitable τ

3 Volume 16 pp April 27 On Roth s Pseudo Equivalence Over Rings Lemmata We begin by deriving some simple consequences of pseudo equivalence Lemma 31 If n m then (i) is symmetric (ii) m is regular if only if n is regular (iii) nr pmr Rn Rmq (iv) mr nr Proof (iv) Let φ(mx) nqx Then nqx 1 nqx 2 mx 1 pnqx 1 pnqx 2 mx 2 Also for any s ns nqq + s φ(mq + s) As such φ is a one-one onto module isomorphism therefore mr nr are isomorphic We may at once apply these to the matrix rings over R state Corollary 32 If M D M M thenr(d) R(M ) The following lemma was proved in 11 characterizes the finiteness of R n Lemma 33 Let e f Rwith e 2 e f 2 f The following conditions are equivalent: 1 R is finite 2 er frer fr er fr 3 Re RfRe Rf Re Rf We shall apply this to regular matrix rings A ey result in our reduction is the following corner lemma Lemma 34 a (a) When yd + then b d+ y a (b) If M r d inverse a (1 dd + )b(1 a + a) d withr (1 dd + )b(1 a + a) then there exists a 1-2 M + a + (1 a + a)r + (1 dd + ) d + such that MM + aa + dd + + rr + (1 dd + ) M + M a + a +(1 a + a)r + r d + d Proof (a) 1 a ba + 1 b d+ y 1 d + b(1 a + a) 1 a (1 dd + )b(1 a + a) d (b) This fundamental result was shown in 9 p 211 eq (35)

4 Volume 16 pp April RE Hartwig P Patrício 4 The cornered canonical forms We next turn to the bloc triangular case Let R be a regular ring let A 1 B 2 A 2 (41) M B A D + B B n be a lower triangular bloc matrix with D D(M) diag(a 1 A n ) We shall assume that A i is p i q j M is p q wherep n p i q n q j Our aim is to address the question of how to characterize M D M D The former was done in 12 with aid of the canonical form A 1 Y 2 A 2 (42) PMQ P (D + B)Q N D + Y Y A A n Y n i1 A n j1 where (43) P P n Pn 1 1 Q Q2 Qn 1 I n 2 1 P2 I n 2 I n π Q n I n P P 1 P2 I 2 P 1 π Q2 Q I 2 1 π Q 1 In these expressions (44) Q I P B π D + 1 I A + B π (I D + D ) 1 I q 1

5 Volume 16 pp April 27 On Roth s Pseudo Equivalence Over Rings 115 are both In addition the submatrices Y of N are defined by (45) Y (1 A A + )B π (I D + D ) The reduction as given in (42) is equivalent to the horizontal reduction (46) V 1 M(U 1 Z)N D + Y in which U I + D + BV I + BD + Z I + D + BDD + are all invertible In order to solve the pseudo-equivalence problem we shall need a second parallel canonical form which again uses Lemma 34 A 1 R 2 A 2 (47) P MQ P (D + B)Q N R A D + R The steps in this reduction are identical to those used to obtain N except that we replace D by the principal bloc N at each stage This gives P P P n P 2 I n I n 2 I n (48) Q Q 2 Q π Q I n 2 I n n I n R n A n (49) P P I P 2 I 2 Q π 2 Q I 2 I Q In these products (41) Q P I B π (N )+ I I A + B π (I (N )+ N ) I I I q

6 Volume 16 pp April RE Hartwig P Patrício are both are It should be noted that π 2 π 2 Q 2 I B 2 A + 1 I 2 2 I A + 2 B 2(I A + 1 A 1) I The submatrices R of N in (47) are defined by (411) R (1 A A + )B π (I (N )+ N ) There does not seem to be an obvious horizontal reduction (using D B () + ) that is equivalent to this total bloc reduction! We next tae advantage of the special form of N Using Lemma 34 we obtain the following Theorem 41 Let N be as in (47) Then there exists a reflexive inverse (N ) + such that N (N ) + diag(a 1 A + 1 A 2A + 2 +R 2R 2 + (1 A 2A + 2 ) A na + n +R nr n + (1 A na + n )) D N Proof IfN then by Lemma 34 we can find a reflexive inverse R A (N ) + (N ) + (I (N ) + N )R+ (1 A A + ) A + Recalling that A + R R (N ) + we may conclude that (N ) (N ) + (N ) (N ) + (412) A A + + R (R ) + (1 A A + ) We note in passing that R D + in general It now follows by induction that if N (N ) + is diagonal then so is N (N ) + has the form N (N ) + diaga 1A + 1 A 2A + 2 +R 2(R 2 ) + (1 A 2 A + 2 ) A A + +R (R ) + (1 A A + ) for 1n When n we arrive at D N n (N ) + n as desired Because the product of A A + I A A + is zero we see that RA A + + R (R ) + (1 A A + ) RA A + RR (R ) + (1 A A + ) as an internal direct sum hence that R(A )R(A A + ) RA A + + R (R ) + (1 A A + )

7 Volume 16 pp April 27 This allows us to conclude that Corollary 42 R(A 1 ) R(D) R(A n ) On Roth s Pseudo Equivalence Over Rings 117 R(A 1 ) R(A 2 ) RR 2 (R 2 ) + (1 A 2 A + 2 ) R(A n ) RR n (R n ) + (1 A n A + n ) We may now combine Corollaries 32 42to conclude that (i) R(D) R(D) (ii) R(M) R(N )RN (N ) + R(D) (iii) If M D then R(D) R(M) R(D) R(D) R(N n) Now if R p p is finite then by Theorem 1 of 1 we may conclude that R(D) R(D) thus RR R + (1 A A + ) This means that R R (R ) + R for 1 2n we have the Roth Consistency Conditions R ie (413) R (1 A A + )B π (I (N ) + N ) In order to relate these conditions to the condition that Y we shall need the following Lemma 43 Let N be as in (47) For each 1 t n the following are equivalent (i) N (N ) + D D + 1t (ii) R 2t (iii) (N )+ N D+ D 1t (iv) Y 2t Proof We shall use induction in all four cases (i) (ii) From Theorem 41 we see that for t N (N ) + (N ) (N ) + A A + + R (R ) + (1 A A + ) If this equals D D + then we must have R (R ) + (1 A A + )for t Postmultiplication by B π (I (N ) + N ) then shows that R (ii) (i) Indeed if N (N ) + D D + then setting R in (412) we see that N (N ) + N (N ) + D D + A A + A A + D D + (ii) (iii) One checs that N 2 + N 2 D 2 + D 2 Next we assume it holds for N r 1 Then from Lemma 34 we see that for N r + + N r r 1 N r 1 A + r A r D + r 1 D r 1 A + r A D r + D r r (iii) (ii) If N + N N + N +(I N + N )R + R A + A D + D

8 Volume 16 pp April RE Hartwig P Patrício then N + N +(I N + N )R + R D + D For 2wehaveD 2 + D 2 N 2 + N 2 thus (I A + 1 A 1)R 2 + R 2 Thisgives R 2 Assuming R for<rshowsthat D r + D r N r + (N N r ) + r 1 (N ) r 1 (I N r 1 + N r 1)R r + R r A + r A r D + r 1 D r 1 +(I D r 1 + D r 1)R r + R r A + r A r This gives (I D r 1 + D r 1)R r + R r hence that R r (ii) (iv) If N + N D + D for t then π π R Y (iv) (ii) If Y 2 then clearly R 2 So assume that R i Y i fori 2 1 Then for these values of i N + i N i D + i D i π i π i Consequently R (I A A + )B π (I N + N )(I A A + )B π (I D + D )Y We may now combine all the above in Theorem 44 Let M be a bloc triangular matrix as in 41 suppose that R p p is finite regular then the following are equivalent: (i) M D(M) (ii) M D(M) (iii) R 2n (iv) Y 2n 5 Bac to the Roth conditions Let us now turn the Roth conditions Y into matrix equations It should be noted that the equivalent condition R isnot so transparent When 2 we see that the first Roth consistency condition becomes which return us to A 2 X YA 1 B 2 R 2 (1 A 2 A + 2 )B 1(I A + 1 A 1) For general we recall the consistency condition 12 (I E)BU 1 (I F ) in which E DD + F D + D Using path products this gives for the (p q) bloc (51) (I A p A + p )(BU 1 ) pq (I A + q A q )

9 Volume 16 pp April 27 where (BU 1 ) pq B pq + r 1 On Roth s Pseudo Equivalence Over Rings 119 ( 1) p>i 1>i 2>>i >q This leads at once to the Roth-matrix consistency condition B pi1 A + i 1 B i1i 2 A + i 2 B i i A + i B i q (52) A p X YA q (BU 1 ) pq p q 1n We may now combine the above with the results of 6 7 Theorem 51 Let R be a regular ring The following are equivalent: (i) R has Roth s pseudo equivalence property ie M D(M) Y (R ) (ii) R has Roth s equivalence property ie M D(M) Y (R ) (iii) R is stably finite Proof (i) (ii) This always holds (ii) (iii) This was shown in 6 (iii) (i) If R is stably finite regular then R N is finite for all N Now if in addition M D(M) then by Theorem 44 we see that the Roth consistency conditions Y holdfor 1n 6 The columnspace case The ey condition in Theorem 41 was that R p is finite This condition can be weaened when q<ptor q beingfinite Todothis we have to repeat the above procedure with column spaces instead of row spaces This time we start from the lower right corner rather from the upper left corner use Lemma 34 to reduce the trailing principal submatrices ˆM Againwehavetwo canonical forms corresponding to the horizontal factorization WV 1 MU 1 N D + Y If we again want to use the matrices D it is more convenient to reverse the numbering of the blocs (61) M A n A n 1 C n C n 1 A 2 C 2 A 1 D + C Even though the new consistency conditions tae a different form from the original Roth conditions we shall show that we actually do get the same canonical matrix M! This will become clear once we identify this reduction with the factorization WV 1 MU 1 M Our aim is to show that the reduction in this case can actually be obtained from the first procedure We need two concepts Definition 61 If A a ij ism n then (i) Ā ā ij where a ij a m+1 jn+1 i (ii) A a ij whereina ij we reverse allproductsifany

10 Volume 16 pp April RE Hartwig P Patrício The former can be thought of as Ā (FAF)T wheref is the flip matrix 1 1 provided we bloc transpose In particular if D diag(a 1 A n )then D diag(a n A 1 ) We shall need Theorem 62 Consider the matrices A m B l C l n over an arbitrary ring Then (ABC) C BĀ Proof ( C BĀ) ij l ( C) iu ( B) uv (Ā) vl whichinturnequals l u1 v1 u1 v1 c l+1 un+1 i b +1 vl+1 u a m+1 j+1 v Now set r l +1 u s +1 v Thisgives ( C BĀ) ij On the other h (ABC) ij l r1 s1 s1 r1 (ABC ) ij c rn+1 i b sr a m+1 js l a is b sr c rj hence s1 r1 l c rj b sr a is Next we have ( (ABC) ) l c ij rn+1 i b sr a m+1 js whichisthe(i j) entryin s1 r1 the RHS We shall apply this to the reduction P MQ N as given in (48)-(411) Theorem 44 ensures that Q M P N in which A n A n 1 N (62) R n Rn 1 A 2 R2 A 1 with R (I N (N ) + ) π C (I A + A ) P In 2 P P In n 2

11 Volume 16 pp April 27 with with π Q On Roth s Pseudo Equivalence Over Rings 121 I 2 1 P 2 P P Q Q n In 2 Q 2 1 Q Q P I 2 To identify this canonical form we recall that In π Q inwhich 1 (N ) + π C I 1 (I N (N ) + ) π C A + I (63) +1 P +1 1I π +1 π I Q +1 Again using Theorem 44 we now obtain π 1 (64) +1 q +1 π +1 In particular Q (65) q n B +1 π D + 1 π q q n 1 1 q D + π C +1 From the form of Q we see that Q I (I DD + ) Q CD + inwhichdd + Q DD + (Here D is labeled bacwards!) This gives Q (I +CD + )I +DD + CD + from which we see that Q (I + DD + CD + )(I + CD + ) 1 Since M D + B D + C (with column partitioning reverse numbering) we can identify these matrices as Q WV 1 wherew I + DD + CD + V I + CD + Lastly if U I + D + C?? then? U U +1 1 D + C +1 U

12 Volume 16 pp April RE Hartwig P Patrício consequently U U 1 D+ C +1 U 1 On the other h we may match this with 64 in which U Note that this uses D + π D +!WehavethusshownthatU 1 P Remars (i) The horizontal reductions V 1 MU 1 Z WV 1 MU 1 respectively correspond to the row column partitioned cases (ii) Since V 1 MU 1 Z WV 1 MU 1 D + Y D + Ȳ weseethatwehavetwo sets of consistency conditions for M D ie Y Ȳ Thatis (66) Y (1 A A + )B π (I D + D ) 2n (67) Ȳ (1 D D + ) π C (I A + A )2n These respectively correspond to the rows or columns of Y being zero In conclusion let us return to Theorem 41 Consider N??? N R as given in 62 We may again apply Lemma 34 together with R A + N + R to obtain a reflexive inverse inverse N such that N + N A + A +(I A + A ) R + R N + N It now follows by induction that if ( N ) + N is diagonal then so is N N has the form N + N diaga + A +(I A + A ) R + R A + 2 A 2 +(1 A + 2 A 2) R 2 + R 2 A + 1 A 1 for 1n When n we arrive at D N + N n n as desired As before because the product of A + A I A + A is zero we see that RSA + A +(1 A + A ) R + R RS(A + A ) RS(1 A + A ) R + R as an internal direct sum hence that RS(A )RS(A + A ) RSA + A +(1 A + A ) R + R )

13 Volume 16 pp April 27 On Roth s Pseudo Equivalence Over Rings 123 This allows us to infer that Corollary 63 RS( D) RS(A 1 ) RS(A n ) RSA + n A n (1 A + n A n) R + n R n RSA + 2 A 2 (1 A + 2 A 2) R + 2 R 2 RS(A + 1 A 1) RS( D) We may now combine Corollaries to conclude that (i) RS( D) RS( D) (ii) RS( M) RS( N )RS N + N RS( D) (iii) If M D then RS( D) RS( M) RS( D) Now if R q q is finite then by Theorem 1 of 1 we may conclude that RS(D) RS( D) thus RS(1 A + A ) R + R This means that R R R+ R for 1 2n we have the dual Roth Consistency Conditions R ie (68) R (I N N + ) π C (1 A + A ) with reverse numbering As before we can use the barred version of Lemma 43 to show that these are equivalent to the simpler consistency conditions that Ȳ of (67) 7 Questions We close with several open questions 1 Can we improve on the Roth-matrix consistency conditions of 52in terms of the blocs of the matrix M? 2 Can we use the above technique to derive consistency conditions for the Stein equation? 3 What is the bloc form of N + N? Can we use induction? 4 Are there any other applications of the bar lemma? 5 What horizontal consistency conditions does the second canonical form correspond to? 6 Can we deduce Theorem 62from the corresponding result with just two factors? Acnowledgment This research was supported by CMAT Centro de Matemática da Universidade do Minho Portugal by the Portuguese Foundation for Science Technology FCT through the research program POCTI We also want to than the referee for suggesting numerous improvements

14 Volume 16 pp April RE Hartwig P Patrício REFERENCES 1 PAraprivatecommunication 2 P Ara KR Goodearl KC O Meara E Pardo Diagonalization of matrices over regular rings Linear Algebra Appl 265: P Ara KR Goodearl KC O Meara E Pardo Separative cancellation for projective modules over exchange rings Israel J Math 15: JK Basalary R Kala The matrix equation AX YB C Linear Algebra Appl 25: A Ben-Israel TNE Greville Generalized Inverses: Theory Applications Wiley New Yor RM Guralnic Roth s theorems decomposition of modules Linear Algebra Appl 39: WH Gustafson Roth s theorems over commutative rings Linear Algebra Appl 23: D Helman Perspectivity cancellation in regular rings J Algebra 48: RE Hartwig Bloc Generalized Inverses ArchRat Mech Anal 61(3): RE Hartwig Roth s equivalence problem in unit regular rings Proc Amer Math Soc 59: RE Hartwig J Luh On finite regular rings Pacific J Math 69(1): RE Hartwig P Patrício R Puystjens Diagonalizing triangular matrices via orthogonal Pierce decompositions Linear Algebra Appl 41: M Henrisen On a class of regular rings that are elementary divisor rings Arch Math 24: D Huylebrouc The generalized inverse of a sum with a radical element: Applications Linear Algebra Appl 246: P Patrício R Puystjens About the von Neumann regularity of triangular bloc matrices Linear Algebra Appl : WE Roth The equations AX YB C AX XB C in matrices Proc Amer Math Soc 3: