Idempotence of finitely generated commutative semifields

Abstract

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.2010 Mathematics Subject Classification. primary: 12K10, 16Y60, 20M14, secondary: 11H06, 52A20.This statement can be now understood as an extension of the folklore theorem referred in the beginning, i.e., that every (commutative) field that is finitely generated as a ring is finite. Of course, the greatest difference is the existence of additively idempotent semifields.Since every semifield is clearly ideal-simple, part (b) of the theorem follows immediately from (c). Also, one can be slightly more precise in the additively constant case: this occurs if and only if there is a finitely generated (multiplicative) abelian group G(·) and the semiring is the semifield S := G ∪ {o}, where o is a new element. Operations that extend the multiplication on G are defined by a+b = o and a·o = o for all a, b ∈ S.

Congruence-simple multiplicatively idempotent semirings

“…In [5,7] additively divisible commutative semirings were studied. Analogous questions may be raised for the multiplicative parts of commutative semirings.…”

“…Remark 5.3. According to [5,Corollary 4.6] if S is a commutative parasemifield, that is finitely generated as a semiring, then the multiplicative semigroup S(•) is finitely generated. Combining this result and 5.1 we see that the semigroup S(•) is idempotent.…”

Semifields and a theorem of Abhyankar

“…A lot of progress has already been made on Conjecture 1.2: building on the results from [9], [10], the conjecture was proved for the case of two generators in [7]. Recently, additively idempotent semifields that are finitely generated as semirings were classified [8] using their correspondence with lattice-ordered groups.…”

Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective

“…Also, 'congruence-and ideal-simple semirings' constitutes another booming area in semiring research which has quite interesting and promising applications in various fields, in particular in cryptography [38] (for some relatively recent developments in this area we refer our potential readers to [4], [5], [21], [22], [23], [30], [31], [32], [39], [40], and [45]). In this respect, in the present paper we consider, in the context of CP-semirings, congruenceand ideal-simple semirings as well.…”