Week graded analogues of Gauss lemma and Eisenstein criterion

This paper continues a series of investigations, devoted to generalized forms of Gauss lemma and Eisenstein criterion. Thus in papers [1] and [2] statements for rings with derivations are given, and in [3] those for $Z$- and $Z^+$-graded rings. In this paper $Z^+$-weak graded rings (which include two previous classes) are considered. Theorem 1 is an analog of Eisenstein criterion, theorem 2 is an analog of Gauss lemma. Some improvement of the result of Kovachich [1] follows from these theorems. Partial necessity of some sufficient conditions introduced in the paper has been demonstrated in theorem 3.