**Ruth Moufang** studied projective planes, introducing Moufang planes and non-associative systems called Moufang loops.

- Both girls attended the Realgymnasium in Bad Kreuznach, Ruth beginning her studies there in 1913.
- Both Ruth and Erica Moufang had considerable artistic skills and this was put to good use in their work on Schwan's geometry book.
- In 1924, Moufang passed her Abitur examination while at the gymnasium in Bad Kreuznach and later that year she began her studies at the Johann Wolfgang Goethe University in Frankfurt am Main.
- Arthur Schönflies was the first full professor at Frankfurt but he retired before Moufang began her studies, being replaced by Carl Ludwig Siegel.
- Other leading mathematicians teaching there when Moufang was a student included Ernst Hellinger, Max Dehn, who was appointed to succeed Ludwig Bieberbach after he left in 1921, Otto Szász, and Paul Epstein.
- Following the award of her doctorate, Moufang was awarded a fellowship which let her spend the academic year 1931-32 undertaking research at the University of Rome.
- During the years that she was teaching at universities, Moufang was also working on her habilitation thesis.
- From 1931 to 1937 she had studied projective planes introducing Moufang planes and non-associative systems called Moufang loops.
- Moufang (1933) showed that another incidence theorem, called the theorem of the complete quadrilateral (or of the invariance of the fourth harmonic point), allows one to introduce coordinates which are elements of an alternating division algebra.
- This and a subsequent paper had the effect of stimulating further research of these algebras and of other nonassociative algebraic structures (Moufang loops).
- Moufang published only one paper on group theory, Einige Untersuchungen fiber geordenete Schiefkörper Ⓣ(Some investigations on a fibre ordered skew field), which appeared in print in 1937.
- Moufang also gives applications of the result to number theory, knot theory and the foundations of geometry.
- Moufang submitted her habilitation thesis in the summer of 1936 and habilitated on 9 February 1937, being only the third German woman to habilitate in mathematics.
- However, the Nazis, to be precise Hitler's minister of education, refused Moufang permission to teach (because she was a woman), so from 1937 she became an industrial mathematician working on elasticity theory.
- This gives Moufang the unique position of being the first German woman with a doctorate to be employed in industry.
- Since there were no permanent positions in universities which consisted of research alone, Moufang left academic life and joined the Krupps Research Institute in Essen where she remained until 1946.
- At this stage in her career Moufang undertook research on applied mathematics topics, in particular on the theory of elasticity.
- In particular the Johann Wolfgang Goethe University in Frankfurt am Main looked to recruit such people and Moufang, having habilitated nine years earlier, was awarded the right to teach there as a docent in 1946.
- Moufang retired in 1970.
- In 2006 a street in Frankfurt was named in her honour and in 2010 the University of Frankfurt set up the Ruth Moufang Fund to support students and scientists from the university.

Born 10 January 1905, Darmstadt, Germany. Died 26 November 1977, Frankfurt am Main, Germany.

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Origin Germany, Women

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive