**Jacob Levitzki**lived for the first eight years of his life in the Ukraine, at that time part of the Russian Empire. In 1912 his family emigrated to Palestine, and Jacob and his brother Asher were educated there. At this time the country was part of the Ottoman Empire and, of the population of around 500,000, about 80,000 were Jews and 70,000 Christians. The family lived in Tel Aviv, at that time a Jewish Community just outside the city of Jaffa. World War I was a difficult time for those in Tel Aviv and, in 1917, the Ottoman authorities evacuated Jews. However, when the war ended in 1918, the Ottomans were defeated and the British took control allowing the inhabitants of Tel Aviv to return to their homes.

Levitzki was educated at the Herzliya Gymnasium in Tel Aviv, the country's first Hebrew school. It was situated on Herzi Street and the building formed an impressive landmark in the town. It was not mathematics that was Levitzki's favourite school subject but rather he loved chemistry and, after graduating from the Herzliya Gymnasium in 1922, he travelled to Germany intending to read for a degree in chemistry at the University of Göttingen.

Benjamin Amirà, also known as Binyamin Amirà, was born in 1896 in what is now the Ukraine and emigrated with his family to Palestine in 1910, two years before the Levitzki family. Like the Levitzki family, the Amirà family lived in Tel Aviv and Benjamin attended the Herzliya Gymnasium. Although Benjamin Amirà was eight years older than Levitzki, the two families were very friendly. After graduating from the Herzliya Gymnasium, Amirà studied mathematics at the University of Geneva. Then Amirà went to the University of Göttingen in 1921 to undertake research for his doctorate advised by Edmund Landau. He was in Göttingen until 1924 so, when Levitzki arrived there to study chemistry in 1922 he was pleased to meet up again with Amirà who persuaded him to attend one of Emmy Noether's lecture courses. Levitzki was captivated by Noether's mathematics and changed from chemistry to study mathematics from that point on. He continued to undertake research at Göttingen advised by Emmy Noether (and also by Edmund Landau) and was awarded his doctorate in 1929 for his thesis *Über vollständig reduzible Ringe und ihre Unterringe* Ⓣ. During his studies at Göttingen he had followed the German tradition and spent one semester at the University of Cologne.

After spending the academic year 1928-29 at the University of Kiel in Germany, Levitzki went to the United States where he was a Sterling Research Fellow at Yale University until 1931. He presented his paper *On normal products of algebras*, written while he was at Yale, to the Society on 3 April 1931 and it was published in the following year. Leaving Yale later in 1931, Levitzki returned to Israel where he was appointed to the Hebrew University of Jerusalem. In fact this was a new university set up in the 1920s and he was aware of much of the foundation process since his friend Amirà, and particularly Amirà's supervisor Edmund Landau, had been closely involved in establishing the Hebrew University of Jerusalem. Landau was a member of the advisory committee of the university, delivered the lecture *Solved and Unsolved Problems in the Elementary Theory of Numbers* in 1925 when the University opened. Landau, with Amirà as his assistant, headed the Institute of Mathematics at the Hebrew University from its opening in 1927. Edmund Landau had left Jerusalem and returned to Germany by the time Levitzki took up his appointment at the Hebrew University, but Amirà was continuing to develop the Institute of Mathematics there along the lines that Landau had established.

Let us quote from [1] regarding Levitzki's research:-

The papers referred to in this quote are all given in our list of Levitzki's papers which we give at THIS LINK.The beginning of his research was in the period when finiteness in algebra was being replaced by chain conditions and the first steps in extending Wedderburn Structure theorems of associative algebras met with the wall of nilpotent elements. It is not surprising that Levitzki's first interest was nilpotent elements and nil subrings. One of his well known result in this area is the nilpotency of the nil subrings of matrix rings and of nil ideals in rings with ascending chain conditions, a result which since has been a source for many generalizations. His work on nilpotent elements played a role in developing the now trivial notion of a rank of a matrix over non-commutative fields and also led to a different notion of rank of elements in I-rings, which was not completely exploited because of his early death.

Levitzki married Charlotte Ascher (9 June 1910 - 7 October 1997), the daughter of Max Ascher and Elsa Gabriele Gumpert. Jacob and Charlotte (known as Lotte) Levitzki had a son, Alexander Levitzki, was born on 13 August 1940. Alexander chose to study chemistry, the subject his father had intended to study before turning to mathematics. Alexander studied at the Hebrew University of Jerusalem and the Wiseman Institute of Science. After a doctorate in Biochemistry and Biophysics and postdoctoral work at Berkeley in California, he joined the faculty of the Wiseman Institute of Science and later the Hebrew University of Jerusalem where he became Professor of Biochemistry.

The Wedderburn structure theorem is a fundamental result proved by Joseph Wedderburn in his 1908 paper *On hypercomplex numbers* which appeared in the *Proceedings* of the London Mathematical Society. In it he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring. Emil Artin, in 1927, generalised this result [1]:-

In fact, Levitzki's paper... to rings with ascending and descending chain conditions on one-sided ideals left open the problem of necessity of both conditions. Only in1938-39did both Hopkins(using left ideals and Levitzki(with right ideals)simultaneously and independently, in two different parts of the world, prove the now classic structure theorems of rings with the descending chain condition. It was unfortunate for Levitzki that his paper was sent to the editor of a European journal and the situation in Germany and the Second World War prevented his work from being presented to the world until much later.

*On rings which satisfy the minimum condition for the right-hand ideals*was published in 1939. It was reviewed by Saunders Mac Lane who wrote:-

These two reviews almost seem to contradict each other, so we will now quote the note that Levitzki appended to his paperThe generalizations of the Wedderburn structure theorems for linear algebras lead to the consideration of rings A satisfying a minimal condition for left ideals(M L I); that is, each non-empty set of left ideals contains an ideal properly contained in no other ideal of the set. For these rings the behavior of the nilpotent elements is of major interest. One problem proposed by Köthe(1930)and treated by Levitzki was recently solved by Charles Hopkins, who proved that in an M L I ring any subring consisting only of nilpotent elements is itself nilpotent[Duke Math. J.4(1938); Ann. of Math.(1939)]. The present paper contains a new proof of Hopkins' result(submitted after publication of Hopkins' first paper). It contains also certain related theorems; for example, a non-nilpotent M L I ring A is a direct sum of a nilpotent left ideal and certain minimal potent left ideals. ... Levitzki's proof is longer than Hopkins', but will later be applied to further classes of rings.

*On rings which satisfy the minimum condition for the right-hand ideals:-*

Levitzki did indeed produce a further generalisation of the Wedderburn structure theorem in his paperThe present note was sent for publication in October1938. In December1938a note by Charles Hopkins was published "Nilrings with minimum condition for admissible left ideals"(Duke Math. J.4(1938),664-667)in which some of the main results of the present note are proved by a different method. Nevertheless I trust that the present note might be still of some interest since the method used here can be applied also to other interesting classes of rings as I hope to show in a following communication.

*On the radical of a general ring*(1943). Saunders Mac Lane, in the above review, mentions questions posed by Gottfried Köthe. Another of these questions was whether the maximal condition on the right ideals of a ring insures the nilpotence of nil-ideals. Levitzki proved this to be the case and indeed proved an even stronger result in his paper

*Solution of a problem of G Koethe*(1945). Let us note that Koethe was an Austrian mathematician who, like Levitzki, began his university career studying chemistry and changed to mathematics.

Shimshon Avraham Amitsur (1921-1994), the author of [1], was a doctoral student of Levitzki's at the Hebrew University being awarded a Ph.D. in 1950. He wrote about Levitzki as a lecturer in [1]:-

In 1954 Levitzki and Amitsur were each winners of the first Israel Prize for Exact Sciences:-Levitzki was known as an excellent teacher, and he had a feeling for advances of modern mathematics. This led him to introduce vector spaces, modules and linear transformation in the first year courses(undergraduate)in algebra at the Hebrew University as early as1946. His students heard him saying that these notions and notations will prevail in modern mathematics and the young generation of mathematicians must encounter them as early as possible - and this he turned into practice.

Let us record that Levitzki's son, Alexander, won the Israel Prize for Life Sciences in 1990. Alexander Levitzki established the Levitzki Prize in memory of his parents Jacob and Charlotte Levitzki. The prize is awarded by the Israel Mathematical Union every two years:-... for their work on the laws of noncommutative rings.

The first award was made in 2009 to Nir Avni of the Hebrew University.... to a young Israeli mathematician for research in algebra or related areas.

**Article by:** *J J O'Connor* and *E F Robertson*

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