Notes On Congruences Over Commutative Semirings

In the paper,we mainly investigate congruences over commutative semirings.Firstly,relationships between ideals and congruences over commutative semirings are discussed and the equivalent conditions of the maximum congruence are given.Secondly,we introduce the quasi-maximal congruence,besides,it is known that the congruence is an equivalent characterization of the quasi-maximal congruence.Next,we define additive prime congruence on a semiring by analogizing subtractive ideal and obtain the equivalent characterization and some properties.At the same time,we also generalize the extension and limitation of the ideal on a ring and gain some conclusions of the extension and limitation of congruences.Finally,some properties about the radicals of congruence and results of the irreducible congruence under chain condition over idempotent commutative semirings are obtained.