Some Studies Of Congruences Over Semirings

As we all know,congruence and ideal are one-to-one correspondence on the ring,and congruence plays an important role in the study of algebra.In this paper,we introduce prime congruence and primary congruence on the semiring by analogy ideal on the ring.We study the properties of prime congruence and primary congruence over commutative additive idempotent semirings,and the main results are as follows.1.We study congruences on commutative additive idempotent semirings.And we give some results on prime congruences and the radical of congruences.Also,we reveal the relation between the radical of a congruence and the set of the prime congruences containing .2.We study primary congruences on commutative additive idempotent semirings,and give the sufficient and necessary conditions of primary congruences from different angles.We define -primary congruence and obtain some results of their structure.3.We define primary decomposition of congruence on commutative additive idempotent semirings and obtain some results of their properties.We obtain the uniqueness theorem by combining -primary congruence.