Ideals of a Bourne Factor Gamma Semiring
Tilak Raj Sharma , Shweta Gupta
Department of Mathematics, H.P.U. Regional Centre Khaniyara, Dharamshala, District Kangra(HP)-176218
*Corresponding Author E-mail:trpangotra@gmail.com, shwetagupta0303@gmail.com
ABSTRACT:
In this paper, the concept of zero divisor,
additively and multiplicatively cancellable elements, austere and 
closure
of a 
semiring,
ideals and ideals
are introduced and their properties are studied.These results will be used to
obtain some other new results regarding bourne factor of asemiring,austere
and closure
of a semiring.
KEYWORDS:semirings,
ideals,
austere semiring,
additively and multiplicatively cancellable and closure
of a semiring.
INTRODUCTION:
The concept of semiring
was introduced by M.M.K. Rao in 1995 [5] as a generalization of semiring as
well as ring
(it may be recalled here that the notion of 
was
first introduced in algebra by N. Nobusawa in 1964). The concept of a semiring
and its sub-
semirings
with left (right) unity was studied by J. Luh [7] and M.M.K. Rao in [5]. The
ideals,
ideals
and 
ideals
of a semiring
were extensively studied by S.Kyuno [6] and M.M.K.Rao [5]. It is well known
that in the theory of semirings,
the properties of their ideals, prime ideals, semiprime ideals and their
generalization plays an important role in their structure theory, however the
properties of an ideal in a semiring and semiring
are somewhat different from the properties of the usual ring ideals.
In order to amend this gap, the concept of ideals
and ideals
in a semiring were first considered by D.R La Torre in 1965 [4]. It is noted
here that the theory of semiring
has been enriched with the help of operator semirings of a semiring
by Dutta and Sardar[8]. Here, we first study the concept of zero divisor,
additively and multiplicatively cancellable elements, austere and closure
of a semiring
and then obtain few of results characterizing ideals of bourne factor of a semiring.
REFERENCES:
1. J. S. Golan, Semirings and their applications, Kulwer Academic Publishers, 1999.
2. J. S. Golan, The Theory of semirings with applications in Mathematics and theoretical computer science .Pitman Monograph and Surveys in pure and applied Mathematics, 1992.
3. H. Hedayati and K.P. Shum, An introduction to Gamma Semirings international journal of algebra, vol. 5, 2011, no. 15, 709-726.
4.
D.R.
laTorre, on 
ideals
and 
ideals
in hemirings, publ. Math. debrecen,129(1965), 219-226.
5. M. M. Krishana Rao, Gamma Semiring-I; Southeast Asian Bull. of Math., Vol. 19 (1995), 49-54.
6. S. Kyuno, On prime gamma rings, Pacific J. Math. 75(78) 185-190.
7.
J.
Luh, On the theory of simple 
rings,
Michigam Math. J. 16 (1969), 65-75.
8.
T.
K. Dutta, S.K. Sardar, On the Operator semiring of a
semiring,
Southeast Asian Bull. Math., springer Verlag, 26(2002), 203-213.
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Received on 20.11.2016 Modified on 27.11.2016 Accepted on 04.12.2016 ŠA&V Publications All right reserved DOI: 10.5958/2349-2988.2017.00028.6 Research J. Science and Tech. 2017; 9(1):171-174.
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