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"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

Open problems of diagonalizability for matrices over rings with finite stable range

We investigation the open problems of diagonalizability for matrices over rings with nite stable range.

The theorem that each matrix over a ring R is equivalent to same diagonal matrix was proved when R is a ring of integers in 1961 by H. Smith [1]. It was gradually extended by Dikson [2], Wedderburn [3], Van der Warder [4] and Jacobson [5] to various commutative and noncommutative Euclidean domains and commutative principal ideal domains and then to noncommutative principal ideal domains by O. Teichmuller [6]in 1937 (then in somewhat sharper form in Asano [7] and Jacobson [8]). The theorem is also known for arbitrary principal ideal rings [9]. Following Kaplansky [10] a ring R is said to be an elementary divisor ring if and only if for every matrix A over R there exist invertible matrices P, Q such that PAQ = D = (dij) - a diagonal matrix, where diiR ∩ Rdii ⊇ Rdi+1R.

In the same work Kaplansky showed that any nitely presented module over elementary divisor ring is direct sum of cyclic modules.

Recall the R - module M is nitely presented if there exists an exact sequence

Rn → Rn → M → 0

which means that M is not only nitely generated, but also the module of "relation between the generators of M" is nitely generated.

In a case of commutative ring converse statement was proved: if nitely presented modules over ring is decompose in a direct sum of cyclic modules, then such ring is a elementary divisor [11]. This result is a partial solution of problem of Wareld [12].

 

1. Problem of Warfield. On which rings every nitely presented module decompose in a direct sum of cyclic submodules. So, for a commutative rings the problem of Warfield equivalent to a problem of describing elementary divisor ring.

 

2. Problem. Problem of a full describing of elementary divisor rings [10, 13].

Particular attention will be devoted to the diagonal reduction of a matrix consisting of a single row or column. If every 1×2 matrix admits diagonal reduction we shall call R a right Hermite ring; if 2×1 matrices admits diagonal reduction, R is a left Hermite ring and both conditions hold, R is a Hermite ring. It is easy to see that a Hermite ring is a Bezout ring (a ring is a Bezout ring if every nitely generated 1-sided ideal is prinsipal). Examples that either implication is risible are provided by Gillman and Henriksen in [14]. In [15] Amitsur proved that a Bezout domain is Hermite.

It is the open problem

 

3. Open problem [13]. Is every a commutative Bezout domain elementary divisor domain?

The notion of stable range come to ring theory from K-theory. This is very useful for solving of many open problem from rings theory. The denition of stable range goes as follows.

We say that a positive integer n is the stable range of a ring R (or, more informally, that R "has stable range n") if whenever

a1R + ... + amR + am+1R = R

for m ≥ n,  ai ∈ R,  i = 1, 2,...,m + 1 there exist elements x1,..., xm ∈ R that

(a1 + am+1x1)R + ... + (am + am+1xm)R = R.

The author showed, that a commutative Bezout ring is Hermite ring if nd only if the stable range of a ring is equal 2 [16].

It is the open problem

 

4. Open problem. Is exist a commutative Bezout ring of nite stable range more than 2?

 

5. Open problem. Is exist a regular ring of stable range more than 2?

Recall that a ring R is called a regular ring, if for every a ∈ R there exist x ∈ R such that axa = a. It is necessary to note, that known examples of simple regular ring has stable range 1 or ∞.

 

6. Open problem. Is exist a simple regular ring of nite stable range more than 1?

Menal and Moncasi proved that a stable range of a right (left) Hermite ring is equal 2 [17].

The author showed that a right (left) Bezout ring of stable range 1 is a right (left) Hermite ring [16]. It is the open problem.

 

7. Open problem. Is a right (left) Bezout ring of stable range 2 a right (left) Hermite one?

Investigating regular ring of stable range 1 (= unit regular ring) Henriksen showed, that any matrices over such ring equivalent to a diagonal matrices [18]. In the some work Henriksen put the open problem.

 

8. Open problem. Problem of a full describing of regular ring over which any matrix equivalent to a diagonal matrix.

Menal and Moncasi proved [17] that over regular ring any matrix equivalent to a diagonal matrix if and only if it is a Hermite ring. It is the open problem.

 

9. Open problem. Is a regular ring of stable range 2 the Hermite one?

All the same time appeared, that whole classes of regular ring are exist, for example, class of separative regular ring [19] over which just a square matrices are equivalent to a diagonal. It is the open problem.

 

10. Open problem. Problem of a full describing ring over which just a square matrices are equivalent to a diagonal matrices.

In appliquing task the most comfortable are invertible matrices exactly when it could be presented in a form of nite product elementary matrices. It is actually problem about investigation of ring in which every invertible matrix be presented in a form of nite product elementary matrix. Such type of investigation were present by Cohn [20]. This investigation showed, that exist rings, which generalize Euclidean ring and which is a ring over which any matrix leading to a canonical diagonal form by elementary transformation. It is the open problem.

11. Open problem. Problem of a full describing of ring over which every matrix admits the diagonal reduction by elementary transformations.

 

References

[1] Smith H.J.S. On systems of linear indeterminate equations and congruences // Philos. Trans. Roy. Soc., London,-1861,-151,2,293-326.

[2] Dickson L.E. Algebras and Their Arithmetics // University of Chicago Press, Chicago, 1923.

[3] Wedderburn J.H.M. Non-commutative domains of integrity // J.Reine Andrew Math.,-1932,-167,1,129-141.

[4] Van der Warder B.L. Moderne Algebra, // -Berlin, New-York, Springer, -1930.

[5] Jacobson N. Pseudo-linear transformations // Ann. of Math.,-1937,-38, 484-507.

[6] Teichmuller O. Der Elementarteilsatz fur nichtkommutative Ringe // Abh. Preuss. Acad. Wiss. Phys.-Math. K1,-1937,-169-177.

[7] Asano K. Neichtkommutative Hauptidealringe. // Act. Sci. Ind. 696, Hermann, Paris,- 1938.

[8] Джекобсон Н. Теория колец. - М., -1947.

[9] Levy L.S. Robson C.J. Matrices and Pairs of Modules // J. Algebra, -1974, -29, -3, -427-454.

[10] Kaplansky I. Elementary divisors and modules // Trans. Amer. Maht. Soc. -1949. -66. -P.464-491.

[11] Larsen M., Lewis W., Shores T. Elementary divisor rings and nitely presented modules // Trans. Amer. Math. Soc. -1974. -187. -P.231-248.

[12] Wareld R.B. Decomposibility of nitely presented modules // Proc. Amer. Math. Soc., -1970, -25, -2, 167-172.

[13] Henriksen M. Some remarks about elementary divisor rings // Michigan Math. J. -1955/56. -3. -P.159-163.

[14] Gillman L., Henriksen M. Rings of continuous functions in which every nitely generated ideal is principal // Trans. Amer. Math. Soc.,-1956,-82, 366-394.

[15] Amitsur S.A. Remarks of principal ideal rings // Osaka Math.Journ. -1963, -15, -59- 69.

[16] Забавський Б.В. Редукція матриць над кільцями Безу стабільного рангу не більше 2 // Укр. мат. журн.  -2003. 55, 4. 550-554.

[17] Menal P., Moncasi J. On regular rings with stable range 2 // J. Pure Appl. Algebra. -1982, -24. - P.25-40.

[18] Henriksen M. On a class of regular rings that are elementary divisor rings // Arch. Math.,-1973,-24, 2, 133-141.

[19] Ara P., Goodearl K., O'Meara K.C., Pardo E. Diagonalization of matrices over regular rings // Linear Algebra and Appl., -1987, -265, -147-163.

[20] Кон П. Свободные кольца и их связи - М.: Мир. -1976. 

 

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About C(X)

B. Zabavsky, 2 Mar, 2018

About C(X)

Bezout rings with finite Krull dimension

A. Gatalevych, 20 May, 2016

Bezout rings with finite Krull dimension

Archive 2012/13

Адміністратор, 1 Feb, 2013

Archive 2012/13

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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
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Scientific seminar, May, 2018