astronomy

Reinhold Baer

Reinhold Baer is a astronomy topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Reinhold Baer rather than just read about it. In short: Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940.

Key takeaways

Reinhold Baer belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Reinhold Baer to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Reinhold Baer from memory before moving on to harder problems.

Reference excerpt

Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings, Baer groups, and Baer subplanes.

Biography Baer studied mechanical engineering for a year at Leibniz University Hannover. He then went to study philosophy at Freiburg in 1921. While he was at Göttingen in 1922 he was influenced by Emmy Noether 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 in 1927. Baer accepted a post at Halle in 1928. There, he published Ernst Steinitz's "Algebraische Theorie der Körper" with Helmut Hasse, first published in Crelle's Journal in 1910. While Baer was with his wife in Austria, Adolf Hitler and the Nazis 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 and Baer accepted. Baer stayed at Princeton University and was a visiting scholar at the nearby Institute for Advanced Study from 1935 to 1937. For a short while he lived in North Carolina. From 1938 to 1956 he worked at the University of Illinois at Urbana-Champaign. He returned to Germany in 1956. According to biographer K. W. Gruenberg,

The rapid development of lattice theory in the mid-thirties suggested that projective geometry should be viewed as a special kind of lattice, the lattice of all subspaces of a vector space... [Linear Algebra and Projective Geometry (1952)] is an account of the representation of vector spaces over division rings, 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 was set up in his honour.

Bibliography

1934: "Erweiterung von Gruppen und ihren Isomorphismen", Mathematische Zeitschrift 38(1): 375–416 (German) doi:10.1007/BF01170643 MR 1545456 1940: "Nilpotent groups and their generalizations", Transactions of the American Mathematical Society 47: 393–434 MR 0002121 1944: "The higher commutator subgroups of a group", Bulletin of the American Mathematical Society 50: 143–160 MR 0009954 1945: "Representations of groups as quotient groups. II. Minimal central chains of a group", Transactions of the American Mathematical Society 58: 348–389 MR 0015107 1945: "Representations of groups as quotient groups. III. Invariants of classes of related representations", Transactions of the American Mathematical Society 58: 390–419 MR 0015108

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 Reinhold Baer at the Mathematics Genealogy Project K.W. Gruenberg & Derek Robinson (2003) The Mathematical Legacy of Reinhold Baer, Illinois Journal of Mathematics 47(1-2) from Project Euclid. Author profile in the database zbMATH 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.

Illustrations

Worked examples

Example 1 — a first encounter with Reinhold Baer

Start with the simplest possible case. Write down what Reinhold Baer claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Reinhold Baer before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Reinhold Baer ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Reinhold Baer

In research: Reinhold Baer appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Reinhold Baer in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Reinhold Baer is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1902 births, 1979 deaths, 20th-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Reinhold Baer outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Reinhold Baer in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Reinhold Baer means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Reinhold Baer out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Reinhold Baer in simple terms?

Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940.

Why does Reinhold Baer matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Reinhold Baer?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Reinhold Baer.