And in the last chapter , we also have the idempotent semiring which satisfies the identity a + ab + a = a + b is an additive normal idempotent semiring , if and only if it is a pseudo - strong right normal idempotent semiring of left zero semirings , and other corollaries 第三章，证明了满足等式a + ab + a = a + b的幂等半环是加法正规的，当且仅当它是左零半环的伪强右正规幂等半环，及相关推论。
In the second chapter , we give the definition of the pseudo - strong right normal idem - potent semiring of v ? semirings . and we have the additive normal idempotent semiring which satisfies the identity a + ab = a + b arises as a pseudo - strong right normal idempotent semiring of left zero semirings 第二章，与第一章平行地构造了v -半环的伪强右正规幂等半环，由这一结构证明了满足等式a + ab = a + b的加法正规幂等半环是左零半环的伪强右正规幂等半环。
And we describe the join r v p of a ring congruence r and an arbitrary congruence p on s . similarly , we discribe all the divisible semiring congruences on a distributive semiring . at last , we give the least distributive lattice congruence on a commutative distributive semiring and an idempotent distributi ve semiring 在第三部分给出一个分配半环上的所有可除半环同余，并且在此半环的满的、闭的、自共轭的理想子半环形成的集合与此半环上的可除半环同余的集合之间建立了一个一一的、保序映射。
First , according to the definition of strong distributive lattice of semirings , we define the pseudo - strong distributive lattice semiring s and the pseudo - direct product , and we prove that the pseudo - direct product that we just define is a semiring , then we prove that s is the pseudo - sub direct product of d and s / 9 首先根据半环的强分配格的定义，定义了伪强分配格半环和一个斗格半环r与任一半环t的伪直积，证明了我们所定义的伪直积是一个半环，并且证明了s是d与s的伪次直积。
