# Modules over Boolean Like Semiring of Fractions

No Thumbnail Available

## Date

2015-11

## Authors

## Journal Title

## Journal ISSN

## Volume Title

## Publisher

Addis Ababa University

## Abstract

The concept of Boolean like rings is originally due to A.L.Foster, in
1946. Later, in 1982, V. Swaminathan has extensively studied the geometry
of Boolean like rings. Recently in 2011, Venkateswarlu et al introduced
the notion of Boolean like semirings by generalizing the concept
of Boolean like rings of Foster. K.Venkateswarlu, B.V.N. Murthy, and
Y. Yitayew have also made an extensive study of Boolean like semirings.
This work is a continued study of the theory of Boolean like semirings
by introducing and investigating the notions; Boolean like semirings
of fractions and Modules over Boolean like semiring of fractions.
A technique of constructing fractions of Boolean like semirings is introduced
and the fractions of Boolean like semirings obtained are precisely
the Boolean like rings of A. L. Foster. In addition, various characterizations
of different classes of ideals (namely, prime, 2-potent prime,
weakly prime, primary, weakly primary, almost primary, semi prime
and 2-absorbing) in Boolean like semiring of fractions are considered
in the sense of extended and contracted ideals in S−1R and in R. In
this case, it has been proved that every ideal of S−1R is an extended
ideal but every ideal of R is not in general contracted. Thus, certain
conditions that amount an ideal of R to be contracted are identified. A
correspondence theorem between certain classes of ideals of R disjoint
from a multiplicative sub set S of R and ideals of S−1R is stated and
proved.
On the other hand, the notion of Modules over Boolean like semirings
is introduced and studied. It is noted that unlike the theory of Modules
in rings, left and right modules structurally found to behave differently
in the sense of getting similar results in both classes. It is shown that
right modules are zero symmetric where as left modules need not be. In
line to this, it is stated and proved that every module over a Boolean
like semiring is a disjoint union of mutually isomorphic zero symmetric
modules. Further, generalizing the results obtained for ideals of R ( in
this dissertation as well as in the works of other authors), certain characterizations
of prime and generalized prime sub modules are studied.
Finally, a method of constructing fractions of modules over Boolean like
semirings is introduced and shown that S−1M is a Boolean like semiring
module over S−1R.

## Description

## Keywords

concept of Boolean like rings is originally