Ideals in Computable Cummutative Rings
by Professor Rod Downey
Abstract: It was only about a hundred years ago that algebra slowly turned away from
an algorithmic approach toward a more abstract, axiomatic approach. The use of computers in understanding algebraic objects has highlighted the need to understand the effective, or algorithmic, content of mathematics.
This talk looks at very basic questions for effectively presented commutative rings with identity and similar questions concerning vector spaces. Here is a basic observation: Every computable such ring that is not a field has a nontrivial ideal. Is this effectively true? What about principal ideals? What about any ideals? We prove that there exist computable rings whose ideals code models of complete extensions of Peano
Arithmetic. As a consequence, we show that the existence of a nontrivial
proper ideal in a commutative ring with identity which is not a field is
equivalent to WKL_0 over RCA_0. We also prove that there are
computable commutative rings with identity where the nilradical is
Sigma^0_1-complete, and the Jacobson radical is Pi^0_2-complete,
I will try to make the notions above make sense. If time permits I will discuss the vector space case.
The first material is joint with Steffen Lempp of the University of Madison-Wisconsin, and Joe Mileti of the University of Chicago. The second paper is joint with these authors plus Denis Hirschfeldt and Antonio Montalban (Chicago) and Asher Kach (Madison).