Simple Commutative Semirings

Toward homological characterization of semirings by e-injective semimodules

“…Congruences on an S-semimodule M are defined in the standard manner, and Cong(M) denotes the set of all congruences on M. This set is non-empty since it always contains at least two congruences-the diagonal congruence [5], a semiring S is congruence-simple iff the only congruences on S are the diagonal △ S and the universal S 2 ; and S is ideal-simple iff S has exactly two ideals (namely 0 and S). Note that these notions are not the same (see, e.g., [28,Examples 3.8]).…”

Semifields and a theorem of Abhyankar

“…, Tn] (for some n depending on the field). We conjecture that his result can not be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings [2], [7]. …”

“…A semiring S(+, ·) is a semifield if moreover S(·) is a group (such a structure is also occasionally called a parasemifield [2], [7]; note that unlike our definition, sometimes a semifield is defined to have a zero element). A semiring is additively idempotent if a + a = a for all a ∈ S.…”

“…In order to classify all ideal-simple (commutative) semirings [2], one needs to study the following conjecture that extends the above-mentioned classical result on fields that are finitely generated as rings:…”

“…Semirings are a very natural generalization of rings that has been widely studied, not only from purely algebraic perspective, but also for their applications in cryptography, dequantization, tropical mathematics, non-commutative geometry, and the connection to logic via MV-algebras and latticeordered groups [2], [3], [4], [5], [6], [15], [16], [17], [18], [19], [20], and [21]. We refer the reader to the aforementioned works for further history and references.…”