A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A po-semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a po-semiring are studied. In particular, it is proved that under some mild assumption the set $Z(A)$ of nonzero zero divisors of $A$ is $A\setminus \{0,1\}$, each prime element of $A$ is a maximal element, and the zero divisor graph $\G(A)$ of $A$ is a finite graph if and only if $A$ is finite. For a po-semiring $A$ with $Z(A)=A\setminus \{0,1\}$, it is proved that $A$ has finitely many maximal elements if ACC holds either for elements of $A$ or for principal annihilating ideals of $A$. As applications of prime elements, it is shown that the structure of a po-semiring $A$ is completely determined by the structure of integral po-semirings if either $|Z(A)|=1$ or $|Z(A)|=2$ and $Z(A)^2\not=0$. Applications to the ideal structure of commutative rings are considered.
Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum
Government College (A), Rajahmundry, A.P., India（拉贾姆恩德里政府学院(A)）；Acharya Nagarjuna University, Guntur, A.P., India（阿查里亚纳加琼纳大学）；Government College For Women(A), Guntur, Andhra Pradesh, India（贡图尔女子政府学院(A)）
Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings
由Hochster引入的谱空间是与交换环的素谱同胚的拓扑空间。在这篇论文中，我们从幂等半环的角度研究了谱空间，幂等半环是一种代数结构，由于它在热带几何中的一些应用而受到了广泛的关注。首先证明了空间是谱的当且仅当它是幂等半环的\emph{prime $k$-spectrum}。事实上，我们通过构造幂等元半环的一个子范畴来丰富Hochster定理，该子范畴与谱空间范畴是反等价的。我们进一步提供了由半环的同余关系的集合所产生的谱空间的例子。特别地，我们证明了幂等半环上的\emph{space of valuations}和\emph{space of prime congruences}是谱的，并且在两者之间存在一个集合的自然双射;这表明了环和幂等半环之间的明显区别。然后我们发展了半环的交换代数的几个方面。我们主要讨论半环的\emph{closure operations}的概念，并给出几个例子。特别地，我们引入了一个\emph{integral closure operation}和一个\emph{Frobenius closure operation}的幂等半环。
