Let A be a commutative ring and I an ideal of A with a reduction Q. In this paper we give an upper bound on the reduction number of I with respect to Q, when a suitable family of ideals in A is given. As a corollary it follows that if some ideal J containing I satisfies J = QJ , then I = QI, where v denotes the number of generators of J/I as an A-module.

BACKGROUND In Kazakhstan, as in many other countries, a transformation of the traditional stereotypes of reproductive behavior, and change the types of family relationships is occurred. It is caused by social status changes, by the transformation of women's valuable orientations and social-role attitudes. The objective of this study was to identify the indicators of reproductive units and the m...

This paper evaluates and ranks the temporal dimensions, known as fourth dimension of urban design, of a number of places in a city by TOPSIS method. TOPSIS method is technique for order preference by similarity to ideal solution. TOPSIS is one of the renowned methods for classical multi-criteria decision-making (MCDM) problems that defines the positive ideal solution and negative ideal solution...