Reinhold Baer (22 July 1902 – 22 October 1979) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, known for his work in
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
. He introduced
injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule ...
s in 1940. He is the eponym of
Baer rings and
Baer group In mathematics, a Baer group is a group in which every cyclic subgroup is subnormal Subnormal may refer to:
*Subnormal body temperature, a common term for hypothermia
* Subnormal operator, a type of operator in operator theory in mathematics
*Sub ...
s.
Biography
Baer studied
mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, ...
for a year at
Leibniz University Hannover
Gottfried Wilhelm Leibniz University Hannover (german: Gottfried Wilhelm Leibniz Universität), also known as the University of Hannover, is a public research university located in Hanover, Germany. Founded on 2 May 1831 as Higher Vocational Sc ...
. He then went to study
philosophy at
Freiburg
Freiburg im Breisgau (; abbreviated as Freiburg i. Br. or Freiburg i. B.; Low Alemannic: ''Friburg im Brisgau''), commonly referred to as Freiburg, is an independent city in Baden-Württemberg, Germany. With a population of about 230,000 (as o ...
in 1921. While he was at
Göttingen
Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
in 1922 he was influenced by
Emmy Noether
Amalie Emmy Noether Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
and
Hellmuth Kneser
Hellmuth Kneser (16 April 1898 – 23 August 1973) was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-mani ...
. In 1924 he won a scholarship for specially gifted students. Baer wrote up his doctoral dissertation and it was published in
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Au ...
in 1927.
Baer accepted a post at
Halle Halle may refer to:
Places Germany
* Halle (Saale), also called Halle an der Saale, a city in Saxony-Anhalt
** Halle (region), a former administrative region in Saxony-Anhalt
** Bezirk Halle, a former administrative division of East Germany
** Hall ...
in 1928. There, he published
Ernst Steinitz's "Algebraische Theorie der Körper" with
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory a ...
, first published in Crelle's Journal in 1910.
While Baer was with his wife in
Austria
Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
,
Adolf Hitler
Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Nazi Germany, Germany from 1933 until Death of Adolf Hitler, his death in 1945. Adolf Hitler's rise to power, He rose to power as the le ...
and the
Nazis
Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right politics, far-right Totalitarianism, totalitarian political ideology and practices associated with Adolf Hit ...
came into power. Both of Baer's parents were Jewish, and he was for this reason informed that his services at Halle were no longer required.
Louis Mordell invited him to go to
Manchester
Manchester () is a city in Greater Manchester, England. It had a population of 552,000 in 2021. It is bordered by the Cheshire Plain to the south, the Pennines to the north and east, and the neighbouring city of City of Salford, Salford to ...
and Baer accepted.
Baer stayed at
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
and was a visiting scholar at the nearby
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
from 1935 to 1937. For a short while he lived in
North Carolina
North Carolina () is a state in the Southeastern region of the United States. The state is the 28th largest and 9th-most populous of the United States. It is bordered by Virginia to the north, the Atlantic Ocean to the east, Georgia a ...
. From 1938 to 1956 he worked at the
University of Illinois at Urbana-Champaign
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Uni ...
. He returned to
Germany
Germany, officially the Federal Republic of Germany (FRG),, is a country in Central Europe. It is the most populous member state of the European Union. Germany lies between the Baltic and North Sea to the north and the Alps to the sou ...
in 1956.
According to biographer K. W. Gruenberg,
:The rapid development of
lattice theory in the mid-thirties suggested that
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
should be viewed as a special kind of lattice, the
lattice of all
subspaces of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
...
'Linear Algebra and Projective Geometry'' (1952)is an account of the representation of vector spaces over
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
s, of projectivities by semi-linear transformations and of dualities by semi-bilinear forms.
He died of heart failure on October 22 in 1979.
In 2016 the
Reinhold Baer Prize
Aracne is an Italian publishing company, founded in 1993 by Gioacchino Onorati and specialized in academic and scientific literature.
It is the only Italian publishing company that does not require exclusive rights for its publications.
Aracne pu ...
for the best Ph.D. thesis in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
was set up in his honour.
Bibliography
* 1934: "Erweiterung von Gruppen und ihren Isomorphismen",
Mathematische Zeitschrift
''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag.
It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erh ...
38(1): 375–416 (German)
* 1940: "Nilpotent groups and their generalizations",
Transactions of the American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 ...
47: 393–434
* 1944
"The higher commutator subgroups of a group" Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...
50: 143–160
* 1945: "Representations of groups as quotient groups. II. Minimal central chains of a group", ''Transactions of the American Mathematical Society 58: 348–389
* 1945: "Representations of groups as quotient groups. III. Invariants of classes of related representations", ''Transactions of the American Mathematical Society'' 58: 390–419
See also
*
Capable group
*
Dedekind group
*
Retract (group theory)
*
Radical of a ring
*
Semiprime ring
*
Nielsen-Schreier theorem
References
* O. H. Kegel (1979) "Reinhold Baer (1902 — 1979)",
Mathematical Intelligencer 2:181,2.
External links
*
* K.W. Gruenberg & Derek Robinson (2003
The Mathematical Legacy of Reinhold Baer ''Illinois Journal of Mathematics'' 47(1-2) from
Project Euclid.
Author profilein the database
zbMATH
zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...
Baer Family's Schedule of 1940 US Census
Reproduction of a talk given by Baer on his last lecture in 1967, before his retirement from the University of Frankfurt hereis a translation.
{{DEFAULTSORT:Baer, Reinhold
1902 births
1979 deaths
Scientists from Berlin
Jewish emigrants from Nazi Germany to the United States
20th-century German mathematicians
Algebraists
University of Freiburg alumni
University of Göttingen alumni
Martin Luther University of Halle-Wittenberg faculty
Princeton University faculty
Institute for Advanced Study visiting scholars
University of Illinois Urbana-Champaign faculty
Goethe University Frankfurt faculty