近预量子数及其在星与半星运算及理想与模系统中的应用
Near prequantales and applications to star and semistar operations and ideal and module systems
摘要：
我们发现，推广的quantales和prequantales理论提供了一个非交换和非联想的抽象理想的理论设置的理论，星操作，半星操作，半素操作，理想系统，和模块系统，相反，后者的理论激励新的结果在quantales和prequantales理论。它的应用包括：预相干预量子和乘法半格的表示定理，简单交换量子和简单乘法格的刻画，预相干预量子上小于给定核的最大有限核的构造，整环上扩展给定星运算的最小半星运算的构造，预相干预量子上最大有限核的构造，预相干预量子上最小有限核的构造，预相干预量子上最小有限核的构造，预相干预量子上最小有限核的构造等。和两个可能有用的素特征非Noether交换环紧闭包的定义。
英文摘要：
We show that a generalization of the theory of quantales and prequantales provides a noncommutative and nonassociative abstract ideal theoretic setting for the theories of star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results in the theory of quantales and prequantales. Applications include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of simple commutative quantales and simple multiplicative lattices; a construction of the largest finitary nucleus smaller than a given nucleus on a precoherent prequantale; a construction of the smallest semistar operation extending a given star operation on an integral domain; and two potentially useful definitions of tight closure for non-Noetherian commutative rings of prime characteristic.
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