Chains of Prime Ideals: Finite and Infinite

Be'eri Greenfeld (Bar-Ilan University )

Wednesday 9th October, 2013 16:00-17:00 Maths 204


As opposed to the commutative case, non-commutative algebras may possess (infinite) chains of prime ideals with non-prime union. We present examples to this phenomenon and discuss several special cases, such as affine algebras with polynomial growth and algebras satisfying a polynomial identity. In the latter case, we prove that the number of non-prime subunions in the same chain is (tightly) bounded; this is far from the general case (e.g. the free algebra contains a chain of primes with infinitely many non-prime subunions).

 An obvious case when all unions over chains of primes are prime is when the algebra satisfies ACC(primes). We introduce the non-commutative version of Hilbert basis theorem: when does ACC(primes) pass to the polynomial ring? If time permits, we introduce several examples and partial results.

 This is based on joint work with Louis Rowen and Uzi Vishne.

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