The multiplicative closed set Sa
Sarab A. Al-Taha · Pure Mathematical Sciences · 2013
Copyright © 2013 Sarab A.Al-Taha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let R be a commutative ring with identity, let a be a none zero element of R, this paper deals with a new definition for a multiplicative closed set Sa = { a ′ε R: a = aa′} of the ring R, we prove some properties of this set. In [1] we give and study the definition of S1 ideal, in [2] we give and study the definition of S2 ideal while in this work we introduce the ring Rsa,the localization of R at Sa. We prove the following result among others, the principal ideal ‹a › is S1ideal of R if, and only if the principal ideal ‹a⁄a′ › is s1 ideal of the ring Rsa. We also prove that the principal ideal ‹a › is S2 ideal of R if, and only if the principal ideal ‹a⁄a′ › is S2 ideal of Rsa.