On Generalized bi-Gamma-Ideals in Gamma-Semigroups

[Abul Basar]

The present day theory of ideals has been standardized in some
respects, and
it is recently being extensively enriched and studied by many
algebraists. This
notion of ideals that was originally formulated by Dedekind for the
ring of
integers of an algebraic number eld, was again generalized by Emmy
Noether
in terms of one-sided and two-sided ideals in associative rings.
Subsequently,
this theory of ideals has won universal acceptance due to its signi
cance in
characterizing dierent algebraic structures as is evident from the vast
litera-
ture available on the topic. Further, Steinfeld[22 and 23] invented
quasi-ideals
of rings and semigroups in 1953 and in 1956 respectively as a
generalization of
one-sided ideals of rings and semigroups and then in 1952, Good and
Hughes
[13] jointly announced the arrival of the bi-ideal of semigroups as a
generalization of one-sided ideals of semigroups. Interestingly, the
concept of bi-ideals
of semigroups was given earlier than the concept of quasi-ideals of
semigroups
and it was subsequently revealed that the bi-ideal of semigroups
generalizes
not only one-sided ideals of semigroups but also quasi-ideals of
semigroups.
Moreover, in 1962, the concept of the bi-ideal was extended to
associative
rings by Lajos [14].
The notion of the generalized biideal[(or generalized (1,1)-ideal] was
rst
introduced in rings in 1970 by Szasz[3 and 4] and then in semigroups in
1984
by Lajos [15, 16, 17, 18 and 19] as a generalization of bi-ideals of
rings and
semigroups. In fact the notion of gamma-semigroups is a generalization
of the con-
cept of semigroups. For any relevant terminologies and unde ned
concepts on
gamma-semigroups in this paper, readers can see [6, 7 and 10]. The
notion of quasi-
gamma-ideals and bi-gamma-ideals in gamma-semigroups was given by
Chinram[11 and 12] in
2006 and in 2007 respectively. The properties of the bi-ideal and the
gener-
alized bi-ideal in semigroups as well as in gamma-semigroups have been
studied by
several authors [1, 2, 8, 9, 20 and 21]. In this paper, we have studied
some gen-
eral classical properties of the generalized bi-gamma-ideal in
gamma-semigroups and also
the prime and irreducible generalized bi-gamma-ideal in
gamma-semigroups. Further, S.
Lajos [17] identi ed a class of semigroups for which some classes of
generalized
bi-ideals are distinct from the class of bi-ideals. He also raised the
problem of
characterizing those semigroups whose generalized bi-ideals are
bi-ideals. This
problem was solved by F. Catino[5]. We have investigated it in
gamma-semigroups.
In fact the class of generalized bi-gamma-ideals of gamma-semigroups is
a generalization of
the class of generalized bi-ideals in semigroups in the same way as
bi-gamma-ideals
in gamma-semigroups are a generalization of bi-ideals in semigroups.
For the current
directions of the theory, readers can refer the reference of this paper
and the
references of the reference.

[Robin Chapman]

On 23/05/2013 12:40, Abul Basar wrote:

> ture available on the topic. Further, Steinfeld[22 and 23] invented

The first of many numbered references in this posting, none of
which are included.

> For the current
> directions of the theory, readers can refer the reference of this paper
> and the
> references of the reference.

Quite...