# Where did Krull dimension and zero-dimensional ideals come from?

I am trying to comprehend an article about primary decomposition of ideals. Zero - dimensional ideals are quite emphasized there. I wonder where zero - dimensional ideals come from, what is the history behind? I found that,in fact $$Krull$$ dimension is zero. But how did Krull invent it? What was he searching for? Sincere thanks.

## 1 Answer

He was concerned with providing an abstract analog to the geometric notion of dimension for algebraic varieties given by polynomial rings. Zero-dimensional ideals are abstract algebraic analogs of discrete collections of points. The idea goes back to Hilbert's conversion of algebraic geometry into the language of rings and ideals, and Dedekind inspired Noether's program of abstracting algebra from the more concrete notions for number and polynomial rings. It was embodied in van der Waerden's Moderne Algebra (1930, based on Artin's and Noether's lectures in 1924-28), and arithmetization of algebraic geometry by Weil and Zariski in 1930-40s.

Krull was influenced by Klein's geometric vision at Erlangen, where he became a professor in 1928, and remarked that "Dedekind's language is better adapted to the needs of arithmetic, but experience shows it seems strange to every beginner". According to McTutor's biography of Krull:

"If Emmy Noether had the greatest influence on the topics which Krull would spend his life researching, it can be seen from this inaugural address that it was Klein who had the greatest influence on Krull's large scale view of mathematics... 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."

"A significant achievement in ring theory was his introduction of what today is called the Krull dimension of a commutative Noetherian ring, and the proof of the principal ideal theorem in this setting. This result was quickly recognised as a decisive advance in Noether’s programme of emancipating abstract ring theory form the theory of polynomial rings... The theorem asserts that if a local ring with unique maximal ideal $$m$$ has (Krull) dimension $$n$$ then $$n$$ is the smallest number such that there are $$n$$ distinct elements of $$m$$ which are not contained in any other prime ideal of the ring. Geometrically, if $$m$$ is the maximal ideal of functions that vanish at a point on a variety, this is the claim that at least $$n$$ functions are needed to pick out a point."

More details can be found in Episodes in the History of Modern Algebra (1800-1950) and McLarty's entry to The Architecture of Modern Mathematics.

• I am so grateful for your answer. Thanks again. Mathematics does not make sense for me if I do not know the history behind. – Tedebbur Jan 24 at 9:44