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 a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

. 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, Baer groups, and Baer subplanes.

Biography

Baer studied

mechanical engineering Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...

for a year at

Leibniz University Hannover Leibniz University Hannover (), also known as the University of Hannover, is a public university, public research university located in Hanover, Germany. Founded on 2 May 1831 as Higher Vocational School, the university has undergone six period ...

. He then went to study

philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

Freiburg Freiburg im Breisgau or simply Freiburg is the List of cities in Baden-Württemberg by population, fourth-largest city of the German state of Baden-Württemberg after Stuttgart, Mannheim and Karlsruhe. Its built-up area has a population of abou ...

in 1921. While he was at

Göttingen Göttingen (, ; ; ) 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. According to the 2022 German census, t ...

in 1922 he was influenced by

Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...

and Hellmuth Kneser. 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 A ...

in 1927. Baer accepted a post at Halle in 1928. There, he published

Ernst Steinitz Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte ( Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...

'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 and ...

, first published in Crelle's Journal in 1910. While Baer was with his wife in

Austria Austria, formally the Republic of Austria, is a landlocked country in Central Europe, lying in the Eastern Alps. It is a federation of nine Federal states of Austria, states, of which the capital Vienna is the List of largest cities in Aust ...

Adolf Hitler Adolf Hitler (20 April 1889 – 30 April 1945) was an Austrian-born German politician who was the dictator of Nazi Germany from 1933 until Death of Adolf Hitler, his suicide in 1945. Adolf Hitler's rise to power, He rose to power as the lea ...

and the

Nazis Nazism (), formally named National Socialism (NS; , ), is the far-right politics, far-right Totalitarianism, totalitarian socio-political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Germany. During H ...

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 and the metropolitan borough of Greater Manchester, England. It had an estimated population of in . Greater Manchester is the third-most populous metropolitan area in the United Kingdom, with a population of 2.92&nbs ...

and Baer accepted. Baer stayed at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

and was a visiting scholar at the nearby

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

from 1935 to 1937. For a short while he lived in

North Carolina North Carolina ( ) is a U.S. state, state in the Southeastern United States, Southeastern region of the United States. It is bordered by Virginia to the north, the Atlantic Ocean to the east, South Carolina to the south, Georgia (U.S. stat ...

. From 1938 to 1956 he worked at the

University of Illinois at Urbana-Champaign The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United States. Established in 1867, it is the f ...

. He returned to

Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...

in 1956. According to biographer K. W. Gruenberg, :The rapid development of

lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...

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 (''p ...

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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

... '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 (or, occasionally, a sfield), 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 multiplicativ ...

s, of projectivities by semi-linear transformations and of dualities by semi-bilinear forms. He died of heart failure on 22 October in 1979. In 2016 the Reinhold Baer Prize 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. History The journal was founded in 1917, with its first issue appearing in 1918. It wa ...

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 pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

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

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 Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent a ...

.
Author profile
in 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 Infrastru ...

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

here
is a translation. {{DEFAULTSORT:Baer, Reinhold 1902 births 1979 deaths Mathematicians from Berlin Jewish emigrants from Nazi Germany to the United States 20th-century German mathematicians Algebraists University of Freiburg alumni University of Göttingen alumni Academic staff of the Martin Luther University of Halle-Wittenberg Princeton University faculty Institute for Advanced Study visiting scholars University of Illinois Urbana-Champaign faculty Academic staff of Goethe University Frankfurt

Biography

Bibliography

See also

References

External links