SMARANDACHE ANTI ZERO DIVISORS
Rebin M. Hassan1*, Suham H.
Awla 2
1Department of Mathematics, College of Basic Education,
University of Raparin, Ranya, Kurdistan Region-Iraq
2Department of Mathematics, College of Education,
Salahaddin University, Erbil, Kurdistan Region-Iraq
Corresponding author email: rebin.mohammed@uor.edu.krd
ABSTRACT:
In this paper, we study and discuss the concept of Smarandache
anti zero divisor (SAZD) element of the ring 
and the group ring 
, where 
is a cyclic group of
order 
generated by 
. Moreover, we introduce and discuss the concept of SAZD ideal
of the ring . Some results related to
the given concepts are proved in detail. Accordingly, a Computer Algebra System (GAP) is
used to verify the results of this study.
KEYWORDS: Zero Divisors, Unit Element, SAZD
Element, SAZD Ideal.
1.
INTRODUCTION
Smarandache
concepts were first introduced by Smarandache (2000). These concepts have been widely
studied by many authors (Padilla, 1998; Srinivas & Rao, 2009; Yongxing, 2005; Kandasamy, 2002a). Kandasamy has published
many books and papers about Smarandache concepts by creating the Smarandache
analogue for the various mathematical theoretical concepts. In 2001, Kandasamy
and 2002, Kandasamy and Chetry introduced Smarandache zero divisor elements in
semigroups, rings, and group rings. A nonzero element 
in a ring 
is called a Smarandache
zero divisor if 
, for some 
, and there exist 
such that
1.
,
od;
if x in B then
break;
fi;
od;
if x in B then
break;
fi;
fi;
od;
if
not x in B then
Add(F,x);
fi;
od;
A:=AsSet(A);
B:=AsSet(B);
F:=AsSet(F);
Print(F,"are
not SAZDs","\n");
Print(B,"are
SAZD","\n");
Print(Size(F),"\n",Size(B),"\n",Size(GR));
REFERENCES
Kandasamy, W. V.
(2001). Smarandache zero divisors. Department of Mathematics Indian Institute
of Technology, Madras.
Kandasamy, W. V.
(2002a). Smarandache Ring. American Research Press.
Kandasamy, W. V.
(2002b). Smarandache Semirings, Semifields, And Semivector Spaces. American Research Press:
Rehoboth, NM, USA.
Kandasamy, W. V.,
& Chetry, M. K. (2002). Smarandache-Zero Divisors in Group Rings.
Department of Mathematics Indian Institute of Technology, Madras.
Padilla, R. (1998). Smarandache
algebraic structures. Bulletin of Pure and Applied Sciences, Delhi, 17(1),
119-121.
Smarandache, F. (2000). Special
Algebraic Structures, in Collected papers. Abaddaba, Oradea, 3, 78-81.
Srinivas, T., & Rao, A. C. S. (2009).
On Smarandache Rings. Scientia Magna, 5(4), 117-124.
The GAP Group (2016).
GAP-Groups, Algorithms and programming, Version 4.8.3. http://www.gap-system.org.
Yongxing, W. (2005).
Research On Smarandache problem in number theory. Hexis, 2, 103-106.