**Wolfgang Krull** proved the Krull-Schmidt theorem for decomposing abelian groups of operators and defined the Krull dimension of a ring.

- Helmuth Krull had a dentist's practice in Baden-Baden and it was in that town that Krull attended school.
- It was the custom in those days for students in Germany to move around various universities during their period of study and Krull was no exception.
- His inaugural address on becoming a full professor at Erlangen was one which says much of how Krull saw mathematics.
- The ten years Krull spent in Erlangen were the most productive period of his career.
- In 1939 Krull left Erlangen to take up a chair at Bonn.
- However, his career was disrupted by the Second World War which began shortly after Krull was appointed to the University of Bonn.
- When his war service had ended in 1946, Krull took up again his post at the University of Bonn and he would remain there for the rest of his life.
- In this final period of his career Krull continued his high level of productivity (he wrote 50 papers in his post-war years in Bonn) and also broadened his mathematical interests.
- Krull's first publications were on rings and algebraic extension fields.
- In 1925 he proved the Krull-Schmidt theorem for decomposing abelian groups of operators.
- In passing from the finite to the infinite case Krull introduced topological ideas.
- In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held.
- Perhaps the reason that the idea of the Krull dimension is such a natural concept is that it encapsulates in an abstract setting the analogues of geometric dimensions.
- Krull carried his work forward, defining further concepts which are today central to modern research in ring theory.
- In 1932 he defined valuations which are today known as Krull valuations.
- He then wrote the remarkable treatise Ideal Theory which remains a beautiful introduction to ring theory but is simply a theory built from the results that Krull had himself proved.
- One could say that Krull had achieved the goal he had in some sense set himself in his Erlangen address and arranged his theory to be self-evident.
- The concept was introduced by Krull in 1938 and his fundamental results were developed into a major theory by mathematicians such as Chevalley and Zariski.
- Indeed much of modern ring theory is still following the path which Krull took, building on the foundations which Emmy Noether had laid.

Born 26 August 1899, Baden-Baden, Germany. Died 12 April 1971, Bonn, Germany.

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Algebra, Origin Germany

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive