Theorem 2 . 2 . 7 s is a direct product of an additive normal c - idempotent semiring and a ring , if and only if 5 " is a pseudo - strong right normal idempotent semiring of left rings 7s是加法正规c一幂等半环和环的直积，当且仅当s是左环的伪强右正规幂等半环定理3
Theorem 3 . 3 s is a " d - idempotent semiring , then s is an additive normal idempotent semiring , if and only if s is a pseudo - strong right normal idempotent semiring of left zero semirings 3s是d一幂等半环，则s为加法正规幂等半环，当且仅当s是左零半环的伪强右正规幂等半环
Theorem 1 . 3 . 3 5 is an a - idempotent semiring , then 5 is a normal idem - potent semiring , if and only if s is a strong semilattice idempotent semiring of rectangular idempotent semirings 定理j设s是人一幂等半环，则s是正规幂等半环，当且仅当s是矩形幂等半环的强半格幂等半环
Theorem 1 . 2 . 9 s is a direct product of a normal a - idempotent semiring and a commutative ring with an identity 1 , if and only if s is a strong right normal idempotent semiring of a - left rings Gs是正规人一幂等半环和含么交换环的直积，当且仅当s是a一左环的强右正规幂等半环
In this paper , we shall characterize the linear operators that strongly preserve nilpotent matrices and that strongly preserve invertible matrices over boolean algebras and antinegative semirings without zero divisors 本文将刻画在布尔代数和非负无零因子半环上强保持幂零矩阵和可逆矩阵的线性算子
And by this we have the structure of the normal idempotent semiring which satisfies the identity ab + b = a + b arises as a strong right normal idempotent semiring of left zero idempotent semirings , and some corollaries 利用这一结构证明了满足等式ab + b = a + b的正规幂等半环是左零幂等半环的强右正规幂等半环，及相关推论。
Furthermore , structure a kind of commutative semirings . namely fractional semiring . then , we prove a universal property of fractional semiring , and we discuss the relations between the ideals of commutative semirings and the ideals of fractional semirings 然后，我们证明了分式半环的泛性质，并且讨论了分式半环与交换半环理想之间的关系
In the second section , we give all the ring congruences on a commutative regular semiring s and show that from the lattice of all full , closed , ideal subsemirings of s to the lattice of all the ring conruences on 5 , there is a lattice isomorphism 首先给出一个交换正则半环上的所有环同余，证明了此半环的所有满的、闭的、理想子半环所形成的格与此半环的环同余格同构。
