خرید و فروخت کے شرعی احکامات، عصرِ حاضر میں اطلاق کی صورتیں: سیرت طیبہ کی روشنی میں

Transactions are part of social life. Sale and purchase are important activities of human life. Islam gives guidance for sale and purchase transactions and the life. Jurists and ulma have addressed the varieties of such transactions as the world became increasingly complex place. This article discuss the basic type of sale and purchase and is being used in Islamic banking system.

Stanley Decompositions and Localization

In Chapter 1, some basic definitions and results from commutative algebra are given. In Chapter 2, we study the behavior of Stanley decompositions under the opera- tion of localization with respect to a variable. We prove how prime filtrations behave under localization. We observe that pretty clean filtrations under localization are still pretty clean filtrations. In Chapter 3, we introduce the concept of Stanley decompositions in the local- ized polynomial ring Sf where f is a product of variable, and show that Sf has a canonical Stanley decomposition and that the Stanley depth does not decrease upon localization. Furthermore it is shown that for monomial ideals J ⊂ I ⊂ Sf , the number of maximal Stanley spaces in a Stanley decomposition of I/J is an invariant of I/J. We also introduce the Hilbert series of Zn -graded K-vector space.