Mykola Komarnytskyi
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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2008 |
| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/153367 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Mykola Komarnytskyi // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859461508551933952 |
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| citation_txt | Mykola Komarnytskyi // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| first_indexed | 2025-11-24T04:41:52Z |
| format | Article |
| fulltext |
Mykola Komarnytskyi was born in 1948 in the village Komarnykyi of
L’viv region. In 1966 he entered the Faculty of Mechanics and Mathemat-
ics of L’viv Ivan Franko University. After graduating from this university
in 1971 he was assigned to the Department of Algebra of the Institute of
Physics and Mechanics in L’viv, wherefrom he was called up for military
service for two years. Returning from the army, Mykola Komartytskyi
worked from 1974 until 1979 as an engineer at the Department of Algebra
of the Institute of Applied Problems of Mechanics and Mathematics did
got at the same time his graduate study at this Institute. At 1979 he got
Candidate of Sciences (Ph.D.) degree.
In February 1979 he was selected as an assistant of the chair of higher
mathematics at the L’viv University. From 1980 he worked as an associate
professor of the chair of geometry, and during 1987 – 1992 he held the
chair of algebra and topology. In the period of 1992 – 1998 Mykola
Komarnytskyi was again an associate professor of the chair of algebra and
topology. During 1993 – 1995 he was at the position of senior researcher
to accomplish his habilitation. In 1998 he earned the degree of Doctor of
Sciences. From 1998 he is a professor of the chair of algebra and topology,
and from 2002 the head of this chair. In 1992 – 1993 and in 1996 – 1999 he
was the deputy dean of the Department of Mechanics and Mathematics.
F Mykola Komarnytskyi
Mykola Komarnytskyi was selected the first chairman of the L’viv Society
of Logicians founded in 1999. He is a member of Editorial Boards of five
scientific journals published in Ukraine and a member of the Council
of Experts on mathematics of the Higher Certification Commission of
Ukraine. He was awarded by the Lviv regional administration prise and
the rank “Excellence of the Education in Ukraine”.
Mykola Komarnytskyi carries on a very considerable scientific and
organizational activity: he manages the City Seminar on algebra, super-
vises and doctoral and candidate theses, works in organizing committees
of many international scientific conferences.
During his work at the Institute of the Applied Problems of Mechanics
and Mathematics and at the Department of Mechanics and Mathematics
of the L’viv National University Mykola Komarnytskyi proved himself
to be an experienced teacher and an active and highly skilled working
researcher. He gave courses in algebra, number theory, discrete math-
ematics, logic, algebra and geometry, as well as various special courses
and special seminars. He was advisor of several Candidate (Ph.D.) theses
(Vovk R.V., 1997; Tushnickiy I.Ya., 1999, Zelisko G.V., 2002, Melnyk I.,
2008). Mykola Komarnytskyi is the author of about 100 scientific publi-
cations, and is highly respected by the international scientific community.
Already being a student, Mykola Komarnytskyi showed the ability
to clearly explain to others what he had learned and his love to mathe-
matics. His students and collaborators consider him as one of the best
lecturers. He is a noted scientist recognized in the world for his con-
siderable contributions to many branches of algebra, and one of leading
experts in algebra and logic in Ukraine.
Mykola Komarnytskyi has considerable scientific achievements in many
areas of modern algebra: theory of rings and modules (especially, the the-
ory of radicals and torsions and in the theory of elementary divisor rings),
model theory, categorical logic. He has got very interesting results about
axiomatization of important classes of rings and modules. This topic was
first studied by Eklöf and Sabbah. They proved in 1971 that the axiom-
atizability of the class of injective modules is equivalent to the fact that
the basic ring is noetherian, and that the classes of semihereditary rings
and Prüfer rings are axiomatizable, while the classes of noetherian rings
and principal ideals rings are not so. Mykola Komarnytskyi investigated
the axiomatization of the so called V -rings, that is of associative rings
R with unit such that all simple left and all simple right R-modules are
injective. In particular, he solved the Cozzens-Faith problem on ultra-
powers of principal ideal domains and described the lattice of left ideals
in an ultraproduct of a family of left Bezout domains.
In addition, he found necessary and sufficient conditions for axiomati-
Mykola Komarnytskyi G
zability of the radical class of any radical in the category of modules over a
Dedekind domain in terms of divisibility of radical modules. Among other
results of Mykola Komarnytskyi in this direction, we should mention the
proof of axiomatizability of the class of noncommutative Prüfer rings (in
collaboration with Ivanna Melnyk). His joint results with G. Zelisko on
the structure of endomorphism rings of ultrapowers of modules and his
description of torsion-theoretical spectra of ultrapowers of a countable
family of principal ideal V -domains play an important role in the ring
theory.
In categorical logic Mykola Komarnytskyi studied several categories
of ring objects in an elementary topos in order to detect the existence of
their model-companions and proved that the category of geometric fields
with supports in the Sierpinski topos has a model-companion, namely,
the theory of algebraically closed objects which are geometric fields.
The problem of diagonalization of matrices is classical, its origin is
the Gauss theorem which states that any matrix over a field is equiv-
alent to the diagonal matrix with 1 and 0 on the diagonal. The first
results on the diagonalization of matrices over integers were obtained by
G. Smith in 1861. He proved that each matrix with integer coefficients
can be reduced by elementary transformations to a diagonal form such
that every diagonal element is a divisor of the next one (such diagonal
form is often called the Smith form). Later the Smith theorem was gen-
eralized to various classes of rings. In particular, Dickson and van der
Warden extended it to certain classes of commutative and noncommu-
tative Euclidean rings, and Teichmüller extended the Smith theorem to
noncommutative principal ideal rings.
All these results had been obtained before Irving Kaplansky intro-
duced the notion of elementary divisor ring, that is such a ring that
every matrix over it can be transformed, multiplying it by invertible ones
(from both sides), to a diagonal matrix such that every diagonal element
is a complete divisor of the next one . I. Kaplansky proved that over an
elementary divisor ring every finitely presented module decomposes into
a direct sum of cyclic modules. For commutative rings the inverse holds,
namely, if every finitely presented module over a ring decomposes in the
direct sum of cyclic modules, then the ring is an elementary divisor ring.
For non-commutative rings this inverse statement is no more true, and
Warfiled stated the problem to find an internal characterization of rings
R such that every finitely presented R-module decomposes into a direct
sum of cyclic submodules.
Mykola Komartytskyi proposed a partial solution to this problem, in-
troducing a new class of rings, called almost invariant elementary divisor
ring. They are such rings that every matrix over it can be transformed
H Mykola Komarnytskyi
to a diagonal form, where all diagonal entries, except maybe the last
one, are invariant elements and each of them is a left divisor of the next
one. Just as in Kaplansky case, one only has to check this property for
1× 2, 2× 2 and 2× 1 matrices. The result on decomposability of finitely
presented modules into direct sums of cyclic ones remains true for such
rings too.
Also Mykola Komarnytskyi investigated the diagonal reduction of
matrices over distributive Bezout domains and showed, together with
B. Zabavsky, that an elementary divisor distributive Bezout domain is a
duo-domain.
Most of known classes of elementary divisor rings depend essentially
on the chain conditions of ideals. The first example of elementary divisor
ring without stabilization of chain of ideals was discovered by Wedder-
burn as early as in 1915; it is the ring of analytic functions. In more
abstract form this example allowed Hellmer to introduce a new class of
elementary divisor rings, which are called adequate rings. Mykola Ko-
marnytskyi showed that if every ideal of a commutative Bezout domain
R is transfinite nilpotent, then R is adequate, so an elementary divisor
ring. He also applied the obtained results to simplification of formulas of
the first order theory of modules.
Mykola Komarnytskyi is also a leading specialist in the theory of dif-
ferential rings. His deep ideas allowed to prove important results about
existence and properties of infinite primary factorizations of ideals and
modules over non-noetherian rings under certain natural restrictions. It
became a starting point of a new general theory of primary factorizations
of differential ideals and submodules of differential modules. Moreover,
the lattices of differential torsions and radicals in the category of the dif-
ferential modules over special differential rings were characterized, and it
allowed to obtain same basic properties of rings of linear differential op-
erators with coefficients in such differential rings; in particular, to prove
that differential operators over differentially closed and universal differ-
ential rings have the Wedderburn property.
Mykola Komarnytskyi also has got interesting results on differential
preradicals and differential pretorsions and torsions in the category of
differential modules over various differential rings, and discovered many
deep properties of such objects. In particular, the conditions under which
differentiations can be extended from a differential module to its bicom-
mutators have been obtained.
At present the scientific interests of Mykola Komarnytskyi concern
model-theoretical characterizations of diferential-injective and differential-
projective modules, properties of ultraproducts of differential rings, a de-
scription of differential spectra of direct products of differentially simple
Mykola Komarnytskyi I
rings. He plans to investigate the differential rings with abelian ring of
Li differentiations, as well as differentiations of certain near-rings, and
to describe properties and the structure of some types ideally differential
rings and modules.
Certainly, we could only mention here a part of scientific achievements
of Mykola Komarnytskyi. He is effectively working in many areas of
modern algebra, and will do still more scientific discoveries.
V. I. Andriychuk, Yu. A. Drozd, V. V. Kirichenko, A. P. Petravchuk,
V. M. Petrychkovych, V. I. Sushchansky, B. V. Zabavsky,
M. M. Zarichnyi
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| id | nasplib_isofts_kiev_ua-123456789-153367 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-24T04:41:52Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | 2019-06-14T03:37:26Z 2019-06-14T03:37:26Z 2008 Mykola Komarnytskyi // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/153367 en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Mykola Komarnytskyi Article published earlier |
| spellingShingle | Mykola Komarnytskyi |
| title | Mykola Komarnytskyi |
| title_full | Mykola Komarnytskyi |
| title_fullStr | Mykola Komarnytskyi |
| title_full_unstemmed | Mykola Komarnytskyi |
| title_short | Mykola Komarnytskyi |
| title_sort | mykola komarnytskyi |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/153367 |