The Semiring of Formal Differences
20th February 2024, 4:00 pm – 5:00 pm
Fry Building, Room 2.04
For rings, there is a natural bijection between (two-sided) ideals and congruences, but for semirings (where we do not require additive inverses), this correspondence breaks. Many basic notions concerning ideals in rings thus split into two different notions for semirings, one for ideals and the other for congruences. There is current interest in extensions of scheme theory to commutative semirings, motivated especially by ideas concerning the "field of one element" in arithmetic geometry. To do so, we must decide if we would rather talk about prime ideals or about prime congruences; if the latter then we must decide what we mean by a prime congruence. Of the several definitions of a prime congruence put forward, the most promising is due to Joó and Mincheva.
In this talk, we will see that the congruences in a semiring R are themselves special ideals in a different semiring, the semiring of formal differences for R. Then Joó and Mincheva's prime congruences are precisely those congruences that are prime ideals in this auxiliary semiring. We will also examine Joó and Mincheva's natural arithmetic of congruences in comparison with classical arithmetic of ideals.
The natural numbers NN is already a rich example for this story. Given time, I will end with an alternative multiplication law for congruences on N that better extends the arithmetic of ideals in Z than Joó and Mincheva's. In particular, this alternative multiplication law distributes over sum of congruences and admits unique factorization of congruences into prime congruences.