"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 finite 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 Wareld [12].

1. Problem of Warfield. On which rings every nitely presented module decompose in a direct sum of cyclic submodules. So, for a commutative rings the problem of Warfield 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 denition 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 fur 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] Wareld 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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