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Message from discussion Proof that A^n = 0 if A is nilpotent

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More options Dec 12 2007, 6:06 am
Newsgroups: sci.math
From: Bill Dubuque <w...@nestle.csail.mit.edu>
Date: 12 Dec 2007 06:06:33 -0500
Local: Wed, Dec 12 2007 6:06 am
Subject: Re: Proof that A^n = 0 if A is nilpotent

Jim Ferry <corkleb...@hotmail.com> wrote:
>On Dec 11 2007, vip.maver...@gmail.com wrote:

>> I would like to request some help if anyone has ideas about
>> how to prove that if A is nilpotent matrix then  A^n = 0?

> If  A : R^n -> R^n, then consider the sequence

>     A^0 R^n,  A^1 R^n,  A^2 R^n,  ...

> Each element of this sequence is a linear space that contains the next
> element.  If the containment is proper, the dimension goes down by at
> least one.  On the other hand, if and when the containment is actually
> an equality, the sequence is constant from that point forward.  This
> constant is 0-d iff A is nilpotent [...]

It recalls a pretty pigeonhole proof of AB=I -> BA=I or, equivalently,
injective (1-1) -> surjective (onto), for finite dim vectors spaces V.
I mentioned this proof here [1] on Apr 24 1999, but forgot to followup
various generalizations, and some references to related literature.

BA = I -> A injective,  since  B  times  Ax = Ay  yields  x = y;  and

A surjective -> AB = I, since for all x exist y: x = Ay = A(BA)y = ABx

Thus  BA = I -> AB = I  reduces to  A injective -> A surjective;  but

LEMMA  A injective  ->  A surjective, for all linear A on finite dim V
where  dim V := maximum length of all subspace chains  R < S < ... < V

PROOF  A injective -> A preserves injections:  R < S -> AR < AS
thus for     R <  S < ... <  V   a subspace chain of max length
its image   AR < AS < ... < AV <= V   would have greater length
if  AV < V,  thus instead   AV  = V,  i.e.  A is surjective.     QED

It's one of the prettiest simple pigeonhole proofs I've discovered.

It fails for infinite dim V; see [1] for some easy counterexamples
employing very simple shift operators inspired by the Hilbert Hotel.
The shift operator is of fundamental importance in linear algebra,
for example see the review of Fuhrmann's book in my prior post [2].

The above proof does not depend upon any specific properties of
vector spaces. In fact, the key lemma is merely a lattice/poset
result that simply says that strict-order-preserving maps cannot
decrease heights. The exact same proof applies to any algebraic
structure V of finite height in its lattice of subalgebras, where
A is any injective endomorphism of V. For example, in the trivial
case (no operations) the structure is just a set and homs are just
set maps, and the proof specializes to a form of the pigeonhole
principle, namely that a one-to-one map on a finite set is onto.
For finite-length modules it yields: injective endomorphisms are
surjective.  Also related are Jordan-Holder results on composition
series (which offer an alternative foundation for dimension theory
of vector spaces). Thus we see that the results  AB = I -> BA = I
and  injective endomorphisms are surjective  are simple consequences
of a general pigeonhole principle for finitely generated structures;
moreover the crucial role of the finiteness hypothesis is now clear.

Of course it is highly unlikely that any of these results are new.
Indeed, a quick AMS Math Reviews search turned up the reviews below
discussing related results regarding finitely generated modules.

--Bill Dubuque
--------------------------------------------------------------------------- ---
45:246  13C05
Vasconcelos, Wolmer V.
Regular endomorphisms of finitely generated modules.
J. London Math. Soc. (2) 4 (1971), 27--32.
--------------------------------------------------------------------------- ---
The author studies the question: When is an injective endomorphism
of a finitely generated module over a commutative ring an isomorphism?

Let  R  be a commutative Noetherian ring, and let  E  be an  R-module of
finite type. Consider the following properties of  f in Hom_R(E,E):
(a) f  is injective;  (b)  f  is right cancellable; (c) f  is left cancellable;
(d) f  is surjective; (e)  f  is an isomorphism.  Then the following relations
hold: (e) <-> (d) -> (c) -> (b) <-> (a).  \O(E) = Hom_R(E,E) admits a two-sided
total ring of quotients which is obtained by localizing \O(E)  at the set of
all non-zero divisors of  E. Consequently, \O(E)  has an Artinian total ring
of quotients if and only if  I, the annihilator of  E , has no embedded primes
and  E  is a torsion-free  R/I-module. In the final section the hypothesis that
R  is Noetherian is dropped. Theorem: Let  E  be a finitely generated torsion
module over a ring whose finitistic global dimension is 1; if  E  has finite
projective dimension, then any injective endomorphism of  E is an isomorphism.
Reviewed by M. Teply
--------------------------------------------------------------------------- ---
41:3460  13.40
Vasconcelos, Wolmer V.
Injective endormorphisms of finitely generated modules.
Proc. Amer. Math. Soc. 25 1970 900--901.
--------------------------------------------------------------------------- ---
The author proves the following theorem concerning arbitrary commutative
rings. Theorem: Any injective endomorphism of a finitely generated  R-module
is an isomorphism if and only if every prime ideal of  R  is maximal.

The proof is elementary and rests on the Cohen-Seidenberg theorem concerning
integral extensions of a ring.
Reviewed by R. E. Mac Rae

Zbl: ... is a well-known exercise that Artinian modules have this property.
http://www.emis.de/cgi-bin/Zarchive?an=0197.31404
--------------------------------------------------------------------------- ---
57:9754  16A52
Armendariz, Efraim P.; Fisher, Joe W.; Snider, Robert L.
On injective and surjective endomorphisms of finitely generated modules.
Comm. Algebra 6 (1978), no. 7, 659--672.
--------------------------------------------------------------------------- ---
The authors discuss the following two problems: (I) [(S]) For what rings are
injective[surjective] endomorphisms of finitely generated modules isomorphisms?
The answer to (I) (also proved by F. Dischinger] is: The matrix rings R_n over
R must be left  \pi-regular (basically d.c.c. on ideals generated by the powers
of an element). This condition is an analog of the condition in the commutative
case that the Krull dimension be zero. In the commutative case, the  R_n 's are
easily shown to be  \pi-regular along with  R: in general this is not known.
The characterization is however strong enough to prove (I) when  R  is a P.I.
\pi-ring or when  R  is perfect. As for (S), the results are less decisive, but
it is shown to be true for P.I.-rings that are integral over the center.
Another class of rings for which (S) is valid is that of von Neumann rings
whose primitive factor rings are Artinian. The paper concludes with various
examples which exhibit a variety of limitations on (S), in particular that
it does not hold for arbitrary P.I.-rings.
Reviewed by W. V. Vasconcelos
--------------------------------------------------------------------------- ---
81k:16015  16A30
Menal, Pere
On  \pi  -regular rings whose primitive factor rings are Artinian.
J. Pure Appl. Algebra 20 (1981), no. 1, 71--78.
--------------------------------------------------------------------------- ---
This paper is concerned with rings  R  that are  \pi-regular (for each  x in
R  there exist a positive integer  n  and an element  y in R  such that
x^n y x^n = x^n  and have the further property that all primitive factor
rings of  R  are Artinian. The main theorem of the paper is that if  R  is
a \pi-regular ring whose primitive factor rings are Artinian, then  R  is
strongly  \pi-regular (for each  x in R  there exist a positive integer  n
and an element  y in R  such that  x^n = x^{n+1}y) and  R  has stable range 1
(for any  a,b in R  with  aR + bR = R , there exists  c in R  such that  a+bc
is a unit in  R ). This theorem leads to the following results concerning
finitely generated modules: If  M  is a finitely generated module over a (von
Neumann) regular ring  R  whose primitive factor rings are Artinian;
consequently,  End_R(M)  is strongly  \pi-regular and has stable range 1, and
M  cancels from direct sums of  R-modules. (The strong \pi-regularity of
End_R(M) had been proven earlier by E. P. Armendariz, J. W. Fisher and R. L.
Snider [Comm. Algebra 6 (1978), no. 1, 659--672; MR 57#9754]. The question of
stable range 1 for End_R(M) and the cancellation property for  M  had been
posed as open problems in the reviewer's book [von Neumann regular rings,
Problem 53, see p. 350, Pitman, London, 1979; MR 80e:16011].) Further results
in this paper deal with a  \pi-regular ring  R  whose index of nilpotence is a
finite integer  n: then the primitive factor rings of  R  are all Artinian of
index at most  n. Also, if  R  is a regular ring of index  n  and  M  is an
m-generated R-module, then  End_R(M) is a \pi-regular ring of index at most mn.
Reviewed by K. R. Goodearl
--------------------------------------------------------------------------- ---
82a:16031  16A64
Hirano, Yasuyuki
Correction to: "On Fitting's lemma" [Hiroshima Math. J. 9 (1979), no. 3,
623--626; MR 80j: 16022].
Hiroshima Math. J.  10 (1980), no. 3, 699.
--------------------------------------------------------------------------- ---
The author provides a revised proof of the implication  (1) -> (2)  in
Proposition 1.  J. W. Fisher had pointed out the error to the author.
--------------------------------------------------------------------------- ---
80j:16022  16A64 (16A30)
Hirano, Yasuyuki
On Fitting's lemma.
Hiroshima Math. J.  9 (1979), no. 3, 623--626.
--------------------------------------------------------------------------- ---
A module  M  satisfies Fitting's lemma if for any endomorphism  f  of  M
there exists an integer  n  such that  M = f^n(M) (+) Ker(f^n). It is shown
that if for every finitely generated  R-module injective endomorphisms are
isomorphisms and surjective endomorphisms are isomorphisms, then every
finitely generated  R-module satisfies Fitting's lemma. This fact is used to
give an alternate treatment of some results of E. P. Armendariz, J. W. Fisher
and the reviewer [Comm. Algebra 6 (1978), no. 7, 659--672; MR 57#9754].
Reviewed by Robert L. Snider
--------------------------------------------------------------------------- ---
92h:16008  16E50 (16P70)
Hirano, Yasuyuki(J-OKAY)
Some characterizations of  \pi -regular rings of bounded index.
Math. J. Okayama Univ. 32 (1990), 97--101.
--------------------------------------------------------------------------- ---
A ring  R  is said to have bounded index  n  if there is some number  n  for
which  a^{n+1} = 0  implies  a^n = 0.  R  is said to be  \pi-regular, if for
any  a  in  R  there is  r  in  R  and  n>0  for which  a^n = a^n r a^n. The
author shows that prime  \pi-regular rings of bounded index are simple
Artinian, thereby enabling him to characterize a ring of bounded index as
being  \pi-regular if and only if every prime factor ring is Artinian. (This
result relies on a theorem of E. P. Armendariz, J. W. Fisher and R. L. Snider
[Comm. Algebra 6 (1978), no. 7, 659--672; MR 57 #9754].) He also shows
that a ring is  \pi-regular of bounded index if and only if the indices of
the endomorphism rings of its cyclic modules are bounded.
Reviewed by Louis Rowen
--------------------------------------------------------------------------- ---
93k:16010  16D80 (13C99 16P99)
Leary, F. C.(1-STBV)
Dedekind finite objects in module categories.
J. Pure Appl. Algebra  82 (1992), no. 1, 71--80.
--------------------------------------------------------------------------- ---
An  R-module  M  is called Dedekind-finite when every injective endomorphism
of  M  is an isomorphism. The author obtains some elementary results on
Dedekind-finite modules and, in particular, on the commutative rings  R  such
that each finitely generated free  R-module is Dedekind-finite. Surprisingly
enough, the relevant papers on the subject, such as those by W. V. Vasconcelos
[Proc. Amer. Math. Soc. 25 (1970), 900--901; MR 41#3460] and E. P. Armendariz,
J. W. Fisher and R. L. Snider [Comm. Algebra 6 (1978), no. 7, 659--672; MR
57#9754], are not even mentioned.
Reviewed by J. L. Gomez Pardo
--------------------------------------------------------------------------- ---
99g:13011  13C13
Barry, M.(SNG-DAKS-MI); Gueye, C. T.(SNG-DAKS-MI); Sanghare, M.(SNG-DAKS-MI)
On commutative FGI-rings.
Extracta Math.  12 (1997), no. 3, 255--259.
--------------------------------------------------------------------------- ---
Let  R  be a commutative ring. An  R-module  M  is said to satisfy property
(I) if every injective endomorphism on  M  is surjective. W. V. Vasconcelos
[Proc. Amer. Math. Soc. 25 (1970), 900--901; MR 41#3460] showed that every
finitely generated  R-module satisfies property (I) if and only if every prime
ideal of  R  is maximal. Call  R  an FGI-ring if every  R-module with property
(I) is finitely generated. It is shown that in a commutative FGI-ring every
prime ideal is maximal and the number of prime ideals is finite. Moreover, a
countable commutative ring is an FGI-ring if and only if it is an Artinian
principal ideal ring.
Reviewed by Daniel D. Anderson