Lying-Over Theorem on Left Commutative Rngs

We introduce the notion of a graded integral element, prove the counterpart of the lying-over theorem on commutative algebra in the context of left commutative rngs, and use the Hu-Liu product to select a class of noncommutative rings. Left commutative rngs were introduced in [1]. I have two reasons to be interested in left commutative rngs. The first reason is that left commutative rngs are a class of commutative rings with zero divisors, and a graduate textbook about commutative algebra can be rewritten in the context of left commutative rngs if prime ideals are replaced by Hu-Liu prime ideals. Hence, studying left commutative rngs is an opportunity of extending the elegant commutative ring theory. Hu-Liu commutative trirings introduced in [2] are a class of noncommutative rings with zero divisors. Because of the first reason and the fact that left commutative rngs are special Hu-Liu commutative trirings, it seems to be true that Hu-Liu commutative trirings are suitable for reconsidering commutative ring theory. Hence, the new notions and ideas appearing in the study of left commutative rngs should be of much benefit to learning about the class of noncommutative rings with zero divisors. This is my second reason of being interested in left commutative rngs. The purpose of this paper is to prove the lying-over theorem on left commutative rngs. My proof is based on the second proof of Theorem 3 on page 257 in [3]. After avoiding the similar arguments, my proof is still much longer than the second proof in [3]. This phenomenon is in fact hardly avoidable in the study of left commutative rngs. Therefore, it is predictable that the new textbook obtained by rewriting a textbook on commutative algebra in the context of left commutative rngs will be much thicker than the textbook on commutative algebra.